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THE   MEASUREMENT 


GENERAL  EXCHAXGE-VALUE 


•■rt^)<^o 


THIil  MKASL'llEMKNl^ 


OF 


GENERAL  EXC^HANGE  VALUE 


BY 


CORREA   MOYLAN    WALSH 


NrU)  gorfe 
THE    MACMILLAN   COMPANY 

LONDON:   JklACMILLAN  &  CO.,  Ltd. 
1901 


Copyright,  1901, 
By  the   MACMILLAN  COMPANY. 


PRESS  OF 
■   ERA  PRINTING  COMPANY, 
LANCASTER.   PA. 


CONTENTS. 


CHAPTER   I. 

THE   NATURE   OF   EXCHANGE-VALUE. 

Section  I.  Four  kinds  of  economic  value,  'i  1.  The  four  kinds  explained. 
I  2.  Their  common  cliaracteristics  and  differentiae.  §  3.  Confusion  of 
opinion  on  the  subject.  .......         pp.   1-6 

Section  II.  The  relativity  of  exchange-value.  ?  1.  Exchange-value,  like 
gravity,  a  relative  quality  and  power  in  things.  Its  causes,  partly  resid- 
ing in  men,  immaterial  in  mensuration.  §  2.  Denial  of  exchange-value  as 
a  quality  in  things  incorrect.  |  3.  Particular  exchange-value  ;  not  the 
quantity  of  another  thing,  nor  a  mere  relation.  §  4.  General  exchange- 
value  :  two  kinds.        ........         pp.   7-13 

Section  III.  The  quantitativeness  of  exchange-value.  ?  1.  Variability  of 
exchange-value — local  and  temporal.  I  2.  Measurement  of  particular 
exchange-values,  g  3.  Equality  between  exchange-values.  ^  4.  Prac- 
tical difficulties  in  determining  particular  exchange-values.  ?  5.  Prac- 
tical difficulties  in  determining  general  exchange-values ;  assistance 
rendered  by  money.  ^  6.  Measurement  of  general  exchange-value  in- 
dispensable          pp.  14-22 

Section  IV.  Mensuration  not  concerned  with  causes.  §  1.  Causes  to  be  in- 
vestigated after  the  measurement  of  variations.  ^,  2.  Errors  of  economists 
on  the  subject.  I  S.  Mensuration  of  exchange-value  not  useless  because 
different  from  mensuration  of  other  kinds  of  value.     .         .         pp.  22-25 

CHAPTER   II. 

THE   CORRELATION   OF    EXCHANGE-VALUES. 

Section  I.  The  correlation  of  the  exchange-values  of  two  things.  ?  1.  The 
variation  of  one  thing  in  another  involves  an  inverse  variation  of  the 
other  in  it.  |  2.  The  mathematical  relations  in  such  variations — recipro- 
cal ratios.  §  3.  The  equality  of  such  variations,  although  the  propor- 
tions are  different.     §  4.   Accompanying  relations  to  other  things. 

pp.  26-32 
V 


VI  CONTENTS 

Section  II.  The  correlation  between  the  exchange-value  of  one  thing  and 
the  exchange-value  of  all  others.  I  1.  The  inverse  variations  of  partic- 
ular exchange-values  involved  ;  and  the  measurement  of  the  variation  of 
the  one  tiling  in  exchange-value  in  all  the  others.  ?  2.  The  variations 
of  the  others  in  general  exchange-value  are  smaller  than  the  variations  of 
the  one.     §  3.  The  same  principles  stated  in  terms  of  money. 

pp.  32-36 

Section  III.  Impossihility  of  one  thing  alone  varying  in  exchange-value. 
§  1.  The  principle  stated  ;  confined  to  exchange-value,  'i  2.  Error  from 
not  recognizing  it  ;  needlessness  of  reasoning  from  probability. 

pp.  36-39 

Section  IV.  Exchange-value  in  all  other  things  and  exchange-value  in  all 
things,  i  1.  TJie  distinction  explained.  ^  2.  Their  differences.  |  2. 
Their  identity  in  certain  cases.         .....         pp.  39-43 

Section  V.  Possibility  of  constancy  of  general  exchange-value.  ?  1.  Two 
conditions.  |  2.  Denial  of  it  under  variations  of  particular  exchange- 
values  ;  consequences.  |  3.  Eefutation  ;  and  demonstration  of  the  pos- 
sibility— by  compensation  in  opposite  variations.  ?  4.  Relations  between 
things  constant  and  things  that  vary.  |  5.  Collectively  all  the  other 
things  may  be  constant  also,      'i  6.  Summary.     ...         pp.  44-53 


CHAPTER   III. 

ox   THE   MEASUREMENT   OF   GENERAL   EXCHANGE-VALUE. 

Section  I.  Comparison  with  other  measurements.  ?  1.  Seeming  peculiar- 
ity of  our  subject  because  of  the  relativity  of  exchange-value.  §  2.  The 
dirterence  really  only  of  degree.        .....         pp.  54-56 

Section  II.  The  standard  in  simple  mensuration.  ?  1.  Need  of  a  standard 
in  every  kind  of  mensuration,  'i  2.  Two  kinds  of  subjects  of  mensura- 
tion. ^  3.  In  subjects  of  simple  mensuration  the  standard  is  the  relation 
between  a  whole  and  its  parts,  'i  4.  The  fixity  neither  in  the  whole  nor 
in  the  parts  singly.  §  5.  We  want  our  measures  to  be  fixed  parts  re- 
latively to  the  universe.  ^  6.  We  practically  content  ourselves  Avith 
reference  to  a  smaller  whole,  'i  7.  And  there  is  need  of  compensatory 
variations  even  in  the  mensuration  of  length  and  weight.  §  8.  The 
mensuration  of  exchange-value  not  essentially  different.  |  9.  The  term 
"aksolute"  can  apply  only  in  a  secondary  sense.         .  .         pp.  56-67 

Section  III.  The  distinction  of  two  kinds  not  peculiar  to  general  exchange- 
value.  ?  1.  The  distinction  likewise  in  the  cases  of  length  and  motion. 
?  2.   Probability  excluded  also  here pp.  67-70 

Section  IV.  The  true  peculiarity  in  the  mensuration  of  general  exchange- 
value.  ?  1.  The  economic  worlds  compared  arc  not  the  same  or  exactly 
alike,  'i  2.  This  difficulty  in  a  less  degree  also  in  other  subjects.  ?  3. 
Two  ways  of  overcoming  it.      ■^  4.   The  amount  of  precision  needed. 

pp.  70-75 


CONTENTS  Vll 

CHAPTER    IV. 

SELECTION'  AND  AMKANTtEMENT  OF  PARTICULAR  EXCHANGt^VALUKS. 

Section  I.  The  Selection.  ^  1.  Need  of  curtailment.  ?  2.  ('lasses  only  to 
be  counted,  and  of  these  the  simplest.  §  3.  Still  further  restriction  for 
measurements  in  the  past.     ^  4.   Only  wholesale  prices  to  be  used. 

pp.  76-80 

Section  II.  Weighting.  |  1.  Definition  of  weighting.  I  2.  Haphazard  and 
even  weighting.  ^  3.  Need  of  care  in  w^eighting.  ^  4.  Various  opinions 
about  weighting;  wiiether  it  should  be  according  to  importance.  ?  5. 
On  weighting  according  to  physical  weight  or  bulk.  .  pp.  80-87 

Section  III.  Weighting  examined  and  explained,  'i  1.  The  nature  of  weight- 
ing. I  2.  Problem  about  the  individuals  in  classes.  ?,  3.  Economic  in- 
dividuals not  equal  weights  or  bulks.  |  4.  They  are  equivalents.  ?  5. 
The  size  of  the  class  the  variation  of  which  in  exchange-value  is  being 
measured,  indifferent.         .         .  .....        pp.  87-95 

Section  IV.  Some  details  in  weighting.  ?  1.  Tho  quantities  not  at  a  mo- 
ment, but  during  a  period.  ^  2.  The  quantity  produced  or  the  quantity 
consumed,  according  to  which  is  the  larger.  ?  3.  Materials  in  different 
stages  of  production.  |  4.  Weighting  also  in  obtaining  the  price  of  each 
class  during  a  given  period.         .         .  ....        pp.  95-97 

Section  V.  The  question  of  periods.  ?  1.  Various  opinions  on  the  subject. 
I  2.  Three  elemental  cases.  |  3.  I.  Classes  with  constant  relative  money- 
valii£s  at  both  periods,  to  be  weighted  according  to  these.  ?  4.  II.  Clares 
ivith  constant  relative  mnss-qufintities  at  both  periods.  The  weighting  of 
neither  period  preferable.  |  5.  Nor  the  weights  that  are  common  to  both 
periods,  being  the  smaller  at  either.  ?  6.  A  mean  must  be  used,  and  this 
the  geometric.  ^  7.  An  example.  §  8.  III.  Classes  with  varying  money- 
values  and  mass-quantities.  Various  positions.  ?  9.  On  the  appearance 
and  disappearance  of  classes;  these  not  to  be  counted.  ?  10.  But  the 
individuals  that  appear  or  disappear  in  old  classes  are  to  be  counted. 
I  11.  Further  rea.sons  against  taking  only  the  lesser  mass-quantities  at 
either  period.  ?  12.  Nor  is  the  arithmetic  mean  of  the  mass-quantities 
proper.  ^  13.  But  the  geometric  mean  of  the  total  money-values.  J.  14. 
Less  exactness  required  in  measuring  variations  in  the  past.     pp.  97-121 

Section  VI.  Exclusion  of  wages  and  earnings  in  general.  ?.  1.  The  claim 
that  the  "price  of  hihor"  should  be  included,  'i  2.  Labor,  not  an  ex- 
changeable object,  has  not  ex(!hange-value.  ?  3.  Labor  the  measure  of 
cost-value,  and  in  a  way  of  esteem-value.  ?  4.  Desirable  also  to  measure 
the  variations  in  the  esteem-value  of  money.  ^  5.  This  by  means  of  the 
average  money-earnings  of  people.  ?  6.  The  measurement  of  the  varia- 
tions in  the  exchange-value  of  money  a  help  also  in  measuring  variations 
in  the  esteem-value  of  all  things  in  general.  |  7.  Absurdity  of  mixing 
the  two  measurements,  one  of  each  of  two  kinds  of  value,  of  money. 
I  8.   It  confuses  also  a  standard  for  one  stratum  of  society  with  a  standard 


Vlll  CONTEXTS 

for  all  people,  'i  9.  Difficulty  about  weighting  wages  and  commodities 
relatively  to  each  other.  |  10.  Neither  the  measurement  of  esteem-value 
nor  the  measurement  of  exchange-value  is  more  properly  the  measure- 
ment of  "value."  §  11.  The  measurements  of  the  different  kinds  of 
value  should  be  kept  distinct pp.  121-135 


CHAPTER   Y. 

MATHEMATICAL   FORMULATION   OF   EXCHANGE-VALUE   RELATIONS. 

Section  I.     Formulation  of  exchange-value  relations  with  even  weighting. 

^  1.  On  the  need  of  measuring  our  measures.  ^  2.  Preliminary  nota- 
tion. ^  3.  Further  notation,  to  express  equations.  ^  4.  Formulation  of 
exchange-value  in  all  other  things, — three  averages.  §  5.  Formulation 
of  the  variation  of  such  general  exchange-value, — still  three  averages. 
§  6.  Another  notation  for  the  formulation  of  variations.  |  7.  Meaning 
of  the  formulae,  and  their  differences.    ....         pp.  136-150 

Section  II.  Formulation  of  exchange-value  relations  with  uneven  weight- 
ing. §  1.  The  previous  formuLe  moditied.  ^  2.  Forniuhe  with  double 
weighting.       .         .         .         .         .         .         .         .         .         pp.  150-156 

Section  III.  Warnings  against  possible  errors.  ^  1.  On  the  variation  of 
averages  and  the  average  of  variations.  ^  2.  The  two  should  agree. 
^1  3,  4.  Possible  errors  in  arithmetic  averaging — perverting  even  weight- 
ing ;  perverting  uneven  weighting.  §  5.  Possible  erroi-s  in  harmonic 
averaging.  |  6.  Possible  errors  in  geometric  averaging.  §  7.  Another 
cause  of  such  errors  :  false  notation.  §  8.  On  the  notion  of  purchasing 
power pp.  157-173 

Section  IV.  Formulation  of  price  relations  with  even  weighting.  ^  1. 
Conversion  of  tlie  preceding  formula  into  formula'  for  prices,  and  the 
relation  between  the  averages.  I  2.  The  simplest  formulae  for  price 
variations.       .........  pp.  173-178 

Section  V.  Formulation  of  price  relations  with  uneven  weighting.  §  1. 
The  last  formula*  jnodified.      ^  2.   Formuhe  with  double  weighting. 

pp.  178-184 

Section  VI.  Possible  errors,  and  errors  incurred.  §  1.  The  criterion,  that 
the  variation  of  the  averages  should  agree  with  the  average  of  the  varia- 
tions. ^  2.  Possible  errors  in  harmonic  averaging.  ^?  3,  4.  Possible  and 
actual  errors  in  arithmetic  averaging: — haphazard  weighting  in  Dutot's 
and  Carli's  methods  ;  in  Scrope's  and  Young's  methods.  ^  5.  Error  of 
Drobisch's  method,  and  its  true  nature.  §  6.  Possible  errors  in  geometric 
averaging.  ^  7.  An  excellence  in  this  averaging  :  Westergaard'  s  argu- 
ment for  it ;  remarks  thereon,  'i  8.  The  proper  procedure  in  a  series  of 
measurements pp.  184-208 

Section  VII.  Formulation  of  exchange-value  in  all  things.  §  1.  Formulae 
for  averaging  variations.     ^  2.   Formulic  for  variations  of  averages. 

pp.  208-211 


CONTENTS  IX 

CHAPTER   VI. 

THE  QUESTIOX  OF  THE  MEANS  AXD  AVERAGES. 

Section  I.  The  problem  and  suggested  solutions.  ?,  1.  The  problem  in  its 
simplest  form.  ^  2.  Some  hasty  solutions  refuted.  ?  3.  Difficulties  in 
the  subject.     ^  4.  The  means  and  averages  suggested  as  solutions. 

pp.  212-218 

Section  II.  History  of  the  question.  ^,  1.  The  unconscious  adoption  of  the 
arithmetic  average.  |  2.  The  question  raised  by  Jevons,  who  adopted 
the  geometric  average.  §3.  General  retention  of  the  arithmetic  aver- 
age. ?  4.  The  harmonic  average  of  price  variations,  and  others.  ^  5. 
Attempted  rejection  of  all  the  averages,  by  employment  of  double  weight- 
ing          pp.  218-225 

CHAPTER   YII. 

BRIEF   COMPARISON   OF   THE  MEANS. 

Section  I.  Another  form  of  the  same  problem.  §  1.  Statement  of  the  new 
form,      'i  2.   Formulation  of  it pp.  226-228 

Section  II.  The  comparison.  ?  1.  Suggestion  of  the  three  answers.  ^  2. 
A  table  of  coiui)ensaturv  variations,  showing  tlieir  chara,cteristics.  |  3. 
Excessiveness  of  the  arithmetic  and  harmonic  answers  ;  moderation  of 
the  geometric pp.  228-232 

CHAPTER   VIII. 

THE   GENERAL   ARGUMENT    FOR   THE   GEOMETRIC   MEAN. 

Section  I.  Equality  in  opposite  variations :  representation  of  it  by  the 
means  and  averages.  ^  1.  Three  possible  ways  of  conceiving  of  equality 
in  opposite  variations  claiming  to  be  compensatory.  ?  2.  These  ex- 
pounded, 'i  3.  Equality  of  distance  traversed.  ^  4.  Formulation  of 
these  equalities  ;  and  tlieir  conditions.  §  5.  These  equalities  exist  in 
complex  as  well  as  in  simple  arithmetic  and  harmonic  variations.  §  6. 
But  tliey  do  not  exist  in  complex  geometric  variations.  Difierentiation 
of  the  geometric  average  from  the  geometric  mean.  |  7.  Connection  be- 
tween the  means  or  averages  and  the  periods  of  weighting.  Equal  op- 
posite variations,  simply,  are  geometric.  §  8.  In  our  subject  the  argu- 
ments for  the  geometric  equality  apply  on  both  sides  of  the  treatment  of 
exchange-value.       ........         pjx  233-245 

Section  II.  Natures  of  the  subjects  to  which  the  means  and  averages  are 
applicable.  ?  1.  Differences  in  regard  to  the  limits  on  the  opposite  sides. 
'i  2.  Our  subject  of  the  kind  for  which  the  geometric  mean  is  suitable. 

pp.  245-2*49 

Section  III.  The  means  and  averages  in  connection  with  exchange-values 
and  prices.     ^  1.    Real  inequality  in  the  arithmetic  and  harniuiiic  varia- 


X  CONTENTS 

tions.  §  "2.  Real  equality  in  the  simple  geometric  variations,  'i  3.  A 
statement  by  Jevons  explained.  §  4.  The  argument  not  applicable  to 
the  geometric  average.      .......         pp.  249-255 

CHAPTER  IX. 

REVIEW   OF   THE  ARGUMENTS    FOR   THE    HARMONIC   AND   ARITHMETIC 
AVERAGES  OF   PRICE   VARIATIONS. 

Section  I.  Analysis  of  tbe  Arguments,  i  I.  The  argument  for  the  harmonic 
average  of  price  variations^f  rom  compensation  by  equal  mass-quantities. 
?  2.  The  argument  for  the  arithmetic  average  of  price  variations — ^from 
compensation  by  equal  sums  of  money.  ^  3.  Comparison  of  these  argu- 
ments.    §  4.  Faultiness  of  the  argument  for  the  harmonic  average. 

pp.  256-262 

Section  II.  Mathematical  relations  ignored  in  these  arguments.  ^  1.  In- 
terconvertibility  of  the  arguments  ;  their  applicability  also  to  the  geo- 
metric mean.  §  2.  Apparently  equal  defectiveness  of  both  the  arguments, 
applied  to  any  of  the  averages.  ^  3.  Laspeyres's  argument  for  the  arith- 
metic average.  ?  4.  Compensation  by  equal  mass-quantities  possible  in 
all  cases  of  opposite  price  variations.  §  5.  Compensation  by  equal  sums 
of  money  also  possible  in  all  such  cases.  .         .         .         pp.  263-274 

Section  III.  Correction  of  the  defects  in  these  arguments.  |  1.  Their  gen- 
eral defect  is  neglect  of  weighting.  §  2.  Also,  neglect  concerning  the 
mass-units  ;  effect  of  this  upon  the  argument  from  compensation  by  equal 
mass-quantities.  Formulation  of  this  argument.  |  3.  Formulation  of 
the  argument  from  compensation  by  equal  sums  of  money.  ?  4.  The 
two  arguments  apply  to  different  conditions,  and  so  are  not  antagonistic. 

pp.  274-282 

CHAPTER    X. 

THE  :N[ETH0D   FOR   CONSTANT  SUMS   OF   MONEV. 

Section  I.     Schematization  of  the  argument  from  compensation  by  equal 

mass-quantities.  ^  1.  The  c'(|uality  not  in  masses  but  in  proportions, — 
three  kinds,  'i  2.  Sciiemata  for  mass-units  equivalent  at  the  iirst  period. 
^  3.  Schemata  for  mass-units  etpiivalent  at  the  second  period.  |  4. 
Schemata  for  mass-units  ecpiivalent  over  both  the  jjeriods.     pp.   283-292 

Section  II.  What  this  argument  really  proves.  ^  1.  The  appearance  of  cor- 
rectness in  tlio  argument  as  used  by  tlie  harmonic  averagist.  'i  2.  The 
fallacy  in  the  argument  for  the  harmonic  average  of  price  variations. 
I  3.  The  fallacy  in  the  argument  a})plied  to  the  arithmetic  average  of 
price  variations.  |  4.  Correctness  of  the  argument  for  the  geometric 
mean  of  price  variations,      'i  5.   Remarks  on  "purchasing  power." 

pp.  293-305 

Section  III.  Formulation  of  the  method,  'i  1 .  Tlie  general  principle,  and 
the  fornuda'  in  reduced  mass-units,      'i  2.   These  forrauhe  extended  and 


CONTENTS  XI 

interpreted,  'i  3.  Formulation  in  ordinary  mass-units,  and  reduction  to 
a  form  of  Scrope's  metliod.  ^  4.  The  method  satisfies  all  the  Proposi- 
tions. ?  5.  Comparison  of  the  formuhp  ;  cases  when  the  geometric  aver- 
age may  be  rigiit pp.  305-313 

Section  IV.  The  deviation  of  the  geometric  average,  'i  1.  Example  with 
tlie  more  important  class  falling  in  price.  ?  2.  Example  with  the  more 
important  class  rising  in  price.  ?  3.  Probable  direction  and  extent  of 
the  error  ;  likelihood  of  neutralization  in  a  long  series.  pp.  313-324 

Section  V.  Testing  of  the  method  in  a  series.  §  1.  Example  with  simple 
harmonic  price  variations.  ^,  2.  Example  with  simple  arithmetic  price 
variations.  §  3.  Neutralization  in  the  harmonic  and  arithmetic  averages 
alternately  used.  ^  4  Disproof  of  them,  and  proof  of  the  geometric 
mean.  ?  5.  Similar  tests  with  uneven  weighting.  ^  6.  Failure  before 
such  tests,  extended  to  four  or  more  periods,  even  of  the  method  here 
advocated.  ^  7.  Explanation  ;  emendation  ;  objection  to  the  emendation. 
^  8.  Extent  of  the  en-or  in  this  method  ;  likelihood  of  neutralization  in 
a  long  series. pp.  324-341 


CHAPTER   XI. 

THE  ^[i:thod  for  coxstant  mass-quantitif.s. 

Section  I.  Schematization  of  the  argument  for  compensation  by  equal  sums 
of  money.  ^  1.  The  equality  properly  in  the  sums,  though  it  may  also 
be  in  the  prices.  |  2.  Schemata  for  mass-units  equivalent  at  the  first 
period.  ?  3.  Schemata  for  mass-units  equivalent  at  the  second  period, 
and  over  both  the  periods.       ......  pp.  342-348 

Section  II.  What  this  argument  really  proves— coincidence  of  the  averages. 
I  1.  Same  (or  similar)  results  given  by  the  three  means  (or  averages) 
of  price  variations,  with  different  weightings.  ^  2.  Therefore  the  argu- 
ment applies  to  all  three,  each  with  its  own  proper  weighting.  |  3. 
Identification  of  the  two  averages  and  one  mean,  properly  weighted,  with 
Scrope's  method.  ?  4.  An  argument  by  transposition  of  prices.  §  5. 
Hence  neither  average  better  than  the  other  or  than  the  geometric  mean 
when  applicable.  |  6.  Consequent  insufficiency  in  the  arguments  made  by 
Jevons  and  Laspeyres pp.  348-355 

Section  III.     Correctness  of   this  argument.     Scrope's  method.     |  1.  The 

reason  for  mistaking  this  argument  as  applicable  only  to  the  arithmetic 
average  of  price  variations.  ^  2.  Proof  of  the  argument.  I  3.  The  gen- 
eral principle,  and  the  formul*  of  the  method.  ^  4.  It  satisfies  all  the 
Propositions.     ^  5.   Correctness  of  the  geometric  mean  of  price  variations 

when  applicable PP-  355-363 

Section  IV.  The  deviation  of  the  geometric  average,  'i  1.  An  example. 
?  2.  Probable  direction  of  the  deviation.  I  3.  Some  extraordinary 
cases pp.  363-368 

Section  V.     Testing  in  a  series.     §  1.    In  a  series  of  three  periods.     ?  2.   In 

series  of  four  or  more  periods.  Absolute  correctness  of  Scrope's  method 
when  the  masses  are  constant.  .....  pp.  368-370 


XU  CONTENTS 

CHAPTER   XII. 

THK   UNIVERSAL   METHOD. 

Section  I.  Two  metliods  suggested.  ?  1.  Possible  extension  of  the  two  pre- 
ceding methods  to  all  cases.     ^2.  Their  proper  forms.      .       pp.  371-374 

Section  II.     A  third  method,  combining  the  two  special  methods.     §  1.  The 

principle  of  tlie  new  nietliod,  and  examples.  §  2.  Development  of  the 
method.  ^,  3.  Its  formulae.  §  4.  It  is  a  method  using  double  weight- 
ing. Comparison  with  Drobisch's  method.  ^  5.  Coraiiarison  with 
Lehr's  method.  §  6.  Test  over  three  periods  of  these  methods  and  of 
Nicholson's.  ^  7.  Test  over  more  than  three  periods  of  the  new  method. 
§  8.  Explanation  of  its  failure  ;  emendation  ;  unsatisfactoriness  of  this. 
I  9.   Another  test  before  which  it  fails pp.  374-396 

Section  III.  Comparison  of  the  three  methods.  |  1.  Testing  of  tlie  geo- 
metric and  Scrope's  emended  methods.  §  2.  Examination  of  the  rem- 
edy, and  of  the  test  itself.  ^  3.  Conditions  of  the  coincidence  of  the 
three  methods.  §  4.  Smallness  of  their  divergence  in  ordinary  cases. 
^  5.  And  their  closeness  to  the  truth.  I  6.  The  choice  between  them  : 
superiority  of  the  emended  form  of  Scrope's  method.  §  7.  Remarks  on 
the  failure  to  find  a  perfect  method.         ....         pp.  396-408 

Section  IV.  Examination  of  other  methods,  and  a  general  test  case.  ^  1. 
Testing  of  Scrope's  method  applied  to  the  arithmetic  means  of  the  mass- 
quantities,  'i  2.  The  conditions  of  this  method  being  correct.  §  3. 
Need  of  reduction  to  these  conditions, — likewise  in  Lehr's  method.  §  4. 
The  reduction  continued  :  the  method  of  approach.  §  5.  Probable  suffi- 
ciency in  practice  of  this  form  of  Scrope's  method.  §^  6-8.  A  complex 
example  : — the  three  superior  methods  applied  to  it ;  other  methods  ap- 
plied to  it ;  rough  methods  applied  to  it.         .         .         .         pp.  409-433 

Section  V.  The  missing  Propositions  supplied.  |  1.  For  cases  with  two 
classes  evenly  weighted.     |  2.   For  cases  M'ith  constant  mass-quantities. 

pp.  433-437 


CHAPTER   XIII. 

THE   DOCTRINE  OF  THE   CONSERVATION    OF    EXCHANGE-VALUE,   AND   THE 
MEASUREMENT  OF   EXCHANGE- VALUE   IN  ALL   THINGS. 

Section  I.  The  doctrine  described.  ^  1.  History  of  the  doctrine.  Need  of 
the  proviso  in  it.      'i  2.   An  empty  form  of  the  doctrine.      .      pp.  438-443 

Section  II.  The  measurement,  and  proof  of  the  doctrine,  when  the  mass- 
quantities  are  constant.  ^  l.*Tlie  measurement,  'i  2.  The  doctrine 
proved.  ^  3.  An  example.  ?^  4-5.  Eelation  between  excliange-value 
in  all  things  and  excliange-value  in  all  other  tilings  ;  and  a  simpler  form 
of  it pp.  443-457 

Section  III.  Results  when  the  mass-quantities  vary.  ?  1.  The  general 
measurement  of  tlie  excliange-value  of  anything  in  all  things.      ^  2.   The 


CONTENTS  XHl 

measurement  of  tlie  variation  of  the  aggregate  amount  of  exchange-value 
in  an  economic  world.  ^  3.  Approximate  relation  between  exchange- 
value  in  all  things  and  exchange-value  in  all  other  things.  §  4.  The 
weight  of  money.  The  commodity  standard.  ^  5.  Measurements  in  this 
standard — to  find  true-prices.  .....  pp.  457-468 

Section  IV.  Need  of  distinguishing  the  two  kinds  of  general  exchange- 
value.  §  1.  An  example  involving  a  seeming  inconsistency,  'i  2.  Solu- 
tion of  the  inconsistency.         ......         pp.  468-471 


CHAPTER   XIV. 

THE  UTILITY   OF  MEASURING  THE  VARIATIONS  IN  THE  EXCHANGE-VALUE 

OF   MONEY. 

Section  I.  The  theoretical  and  practical  purposes.  ?.  1.  Necessary  for  in- 
vestigation into  effects  of  such  variations.  ^  2.  Necessary  for  measure- 
ments of  the  variations  of  commodities  ;  need  of  true-prices.  ^  3.  Neces- 
sary for  operating  the  scheme  to  pay  debts  in  equal  exchange-value. 
§  4.  Necessary  for  operating  the  scheme  of  regulating  the  exchange-value 
of  money.     ^  5.  Warnings  on  these  subjects.  .         .         pp.  472-481 

Section  II.  Principles  involved  in  the  scheme  of  regulating  the  exchange- 
value  of  money.  ^  1.  Relations  between  variations  of  prices  and  varia- 
tions of  exchange-values.  ^  2.  The  desire  for  everything  to  become 
cheaper,  impossible  of  exchange-value.  §  3.  The  desire  for  everything  to 
become  cheaper  in  price  :  nature  of  the  standard  involved, — of  what  kind 
of  value  should  money  be  the  fixed  measure?  |  4.  Principles  of  issuance 
in  order  to  obtain  constancy  of  the  exchange-value  of  money.  ?  5.  Simi- 
lar principles  of  issuance  if  money  is  the  standard  of  cost-value  or  of 
esteem-value.     ^  6.   Imperfect  science  without  measurement. 

pp.  481-495 


APPENDIX    A. 

ON   VARIATIONS   OF   AVERAGES  AND   AVERAGES   OF   VARIATIONS. 

Introduction.  pp.  497-498 

Section  I.    On  averages  in  general.        .....  pp.  498-501 

Section  II.    Arithmetic  averaging pp.  502-506 

Section  III.     Harmonic  averaging pp.  506-510 

Section  IV.     Cases  of  agreement  between  the  arithmetic  and  the  harmonic 
averages  of  variations.  ......         pp.  510-513 

Section  V.     Geometric  averaging pp.  514-516 

Section  VI.     Comparison  of  the  geometric  average  with  the  other  two. 

pp.  516-521 

Section  VII.     Comparisonof  averages  of  unequal  sets.  .       pp.  521-523 


XIV 


CONTENTS 


APPENDIX    B. 

ON   COMPENSATORY   VARIATIONS 
Section  I.     AVith  terms  equal  at  the  first  period. 
Section  II.     With  terms  equal  at  the  second  period. 

Section  III.     Explanations 

Section  IV.     Combination  of  the  two  kinds  of  variations 
Section  V.     Transposition  of  terms.      .  .  .  , 


pp.  524-526 
pp.  526-527 
pp.  527-529 
pp.  529-531 
pp.  531-532 


APPENDIX   C. 

REVIEW    AND    ANALYSIS    OF    THE    ^METHODS    EMPLOYING  ARITHMETIC 

AVERAGING   FOR   MEASURING    VARIATIONS    IN    THE 

EXCHANGE-VALUE   OF    MONEY. 


Introduction. 

Section  I.     Dutot's  method. 
Section  II.     Carli's  method  and  its  varieties. 
Section  III.     Young's  method  and  its  varieties. 
Section  IV.     Scrope's  method  and  its  varieties. 


pp.  533-534 
p.  535 
pp.  535-536 
pp.  536-539 
pp.  539-544 


Section  V.     Methods    employing    double  weighting.     Drobisch's.     Lehr's. 
Nicholson's.     Others.     .......         pp.  544-552 


Bibliography. 


pp.  553-574 


Index. 


pp.  575-580 


ERRATA. 


Page  3,  line  4  in  notes,  before  Del  Commercio  insert  Elementi 

di  economid  pubbUca 

Page  3,  line  8  in  notes,  for  later  read  latter 

Page  9,  line  2 ,  for  had  read  played 

Page  23,  line  19,  for  outer  read  other 

Page  108,  line  20,  before  the  second  insert  at 

Page  136,  line  2  in  note,  before  value  insert  the 

Page  160,  line  10,  for  variation  read  variations 

Page  198,  line  6  in  notes,  for  some  read  same 

P  P 

Page  202,  below,  twice,  for  ^  read  ^ 

Page  271,  in  the  upper  equation,  for  1  +  (3^^  read  1  —  j3/ 

Page  317,  line  3  in  note, /or  argument  read  agreement 

Page  318,  line  10  in  note,  for  Sect.  II.  read  Sect.  III. 

Page  370,  in  the  heading,  for  sums  read  masses 

Page  409,  in  tlie  formula,  for  .iv,  -f-  •'■•i  read  a\  +  ^2 

Page  462,  lines  23  and  24,  for  p  read  j)' 

Page  522,  line  2,  in  the  denominator,  for  71.,  read  n^ 

Page  530,  line  26,  for  variation  read  variations 

Page  535,  line  4,  for  substracted  read  subtracted 

Page  552,  for  l/x^  and  Vy^y.^  read  V a^a.^  and  V^/3,/3, 

Page  557,  line  21,  for  '53  read  '52 


L'' algebra  non  essendo  che  un  metodo  preciso  e  speditissimo  di  ragio- 
nare  sulle  quantity,  non  e  alia  sola  geometria  od  alle  altre  scienze  mafe- 
matiche  che  si  possa  applicare,  ma  si  pud  ad  essa  sotioporre  tutlo  cib 
che  in  qualche  modo  pud  crescere  o  diminuire,  tutto  cib  che  ha  relazioni 
paragonabili  tra  di  loro.  Quindi  anchc  le  scienze  polHiche  possono  fino 
ad  un  certo  segno  ammetterla. 

Beccaria,  1765. 


THE    MEASUREMENT 


GENERAL  EXCHANGE -VALUE. 

CHAPTER   I. 
THE   NATURE  OF  EXCHANGE -VALUE. 

I. 

§  1.  "Value"  is  an  ambiguous  term,  in  the  common  use  of 
which  may  be  detected  several  meanings  that  deserve  to  be 
distinguished  by  special  epithets.  Applying  the  term  to  spe- 
cies of  material  things,  we  often  have  in  mind  their  useful- 
ness or  utility,  as  when  we  speak  of  water  being  very  valuable 
for  mankind  ;  and  therefore  what  we  refer  to  should  be  called 
use-value.  Or  we  may  be  thinking  of  the  particular  things  in 
a  species,  and  whether  the  species  be  important  or  not  as  such, 
yet  because  we  highly  prize  or  esteem  the  particular  things  for 
their  rarity,  Me  attach  also  value  to  them,  and  this  is  some- 
thing which  should  be  characterized  as  esteem-value.  Again, 
we  sometimes  refer  merely  to  the  fact  that  the  things  are  pro- 
duced or  procured  only  by  labor,  and  we  value  them  because 
they  are  endeared  to  us  by  past  effort  and  cannot  be  replaced 
except  at  the  cost  of  more  effort ;  and  then  we  should  call  the 
quality  we  are  thinking  of,  cost-value.  Lastly,  we  may  be  con- 
sidering only  the  fact  that  things  once  possessed  may  be  ex- 
changed for  other  things,  whereby  a  thing  useless  in  itself  to 
its  owner,  which  he  does  not  esteem  for  itself,  and  which  per- 
haps has  come  to  him  gratuitously,  may  procure  for  him  a  use- 
ful and  needed  object,  and  save  him  the  trouble  of  special 
1  1 


2  THE    NATURE    OF    EXCHANGE-VALUE 

effort  to  produce  that  object, — which  attribute  in  things  is 
properly  called  their  exchange-value. 

Having  the  first  meaning  in  mind,  we  say  a  thing  is  more  or 
is  less  valuable  according  as  its  class  is  more  or  is  less  useful  as 
a  whole — its  "  total  utility  "  is  greater  or  smaller — the  whole 
class  is  less  or  more  dispensable — our  first  want  for  it  is  greater 
or  smaller.  Having  in  mind  the  second,  we  say  that  an  indi- 
vidual thing,  along  with  its  mates,  is  more  or  is  less  valuable  ac- 
cording as  it  is  more  or  is  less  prized  by  people  in  general,  which 
is  generally  according  as  their  uses  for  its  class  are  greater  or 
smaller  and  as  the  abundance  of  its  class  proportionally  to 
their  numbers  is  smaller  or  greater,  or  according  to  the  magni- 
tude of  what  has  been  called  its  "  final  utility  " — its  usefulness 
when  satisfying  the  last  want  which  its  class  is  abundant  enough 
to  satisfy.  Thus  when  we  say  "  water  is  more  valuable  than 
diamonds,"  we  are  comparing  the  species  water  with  the  species 
diamond  ;  when  we  say  "  diamonds  are  more  valuable  than 
water,"  we  are  comparing  individual  diamonds  with  equally 
large  individual  drops  of  water.  Having  in  mind  the  third 
meaning,  we  say  a  thing  is  more  or  is  less  valuable  according 
as  there  is  greater  or  smaller  difficulty  for  people  to  produce  it 
by  finding  it  or  by  making  it.  Having  in  mind  the  last,  we 
say  a  thing  is  more  or  is  less  valuable  according  as  it  procures 
in  exchange,  or  purchases,  more  or  less  of  other  things. 

§  2.  A  term  rarely  has  several  meanings  without  some  one 
idea  running  through  them  all.  In  the  case  of  "  value  "  the 
underlying  idea  is  that  everything  which  we  pronounce  "  valu- 
able "  is  an  object  of  desire.  Everything  desired,  however,  is 
useful  to  us.  But  some  species  of  physically  useful  things  are 
so  abundant  that  we  are  not  aware  of  our  desire  for  them  until 
we  stop  to  imagine  ourselves  deprived  of  them.  Then  we  recog- 
nize their  value — their  use-value.  For  other  things  we  know 
our  desire  because  in  their  absence  we  feel  the  want  of  them. 
Thus  the  more  useful  and  the  more  absent  are  they,  the  more 
we  desire  them,  provided  they  are  not  so  entirely  absent  as  to 
leave  us  unacquainted  with  their  utility  ;  and  when  we  possess 
them,  the  more  we  prize,  esteem,  and  value  them,  for  fear  of 


FOUR    KINDS    OF    ECONOMIC    VAIAJE  3 

losing  tliem.  That  avc  desire  things  whicli  cost  us  effort  to 
produce,  is  shown  by  tlie  fact  that  we  spend  effort  to  procure 
them.  And  anything  is  desirable  to  anybody,  however  useless 
in  itself  to  him,  that  serves  the  useful  purpose  of  providing  him 
with  other  useful  things,  and  so  saves  him  the  labor  otherwise 
necessary  to  obtain  them. 

By  reason  of  this  common  essence,  "  value  "  may  be  defined 
as  the  valor — might,  power — in  things  by  which,  in  general,  or 
under  certain  circumstances,  they  are  rendered  objects  of  desire. 
Things  are  objects  of  desire  in  four  different  ways.  They  are 
objects  of  desire  because  of  their  utility  alone ;  wherefore, 
using  them,  we  assign  to  them  use-value.  They  are  objects 
of  desire  because  of  their  utility  and  rarity;  wherefore,  es- 
teeming them  and  holding  fast  to  them,  Ave  assign  to  them  es- 
teem-value. They  are  objects  of  desire  because  of  their  utility 
and  the  difficulty  of  making  or  replacing  them  ;  wherefore, 
laboring  to  produce  them,  we  assign  to  them  cost- value.  They 
are  objects  of  desire  because  of  their  special  utility  in  providing 
us  with  other  desired  objects  ;  wherefore,  accepting  or  laboring 
to  get  even  what  Ave  otherwise  do  not  Avant  in  order  by  ex- 
change to  get  what  avc  do  want,  Ave  assign  to  them  exchange- 
value.  Apart  from  the  common  reference  to  desirableness,  the 
four  meanings  of  "  value  "  are  very  distinct,  so  that  they  repre- 
sent four  kindK  or  species  of  value. ^  The  distinctness  of  these 
is  also  shown  by  the  fact  that  the  same  thing,  compared  Avith 
others,  may  possess  different  degrees  of  use-value,  of  esteem- 
A^alue,  of  cost-A'alue,  and  of  exchange-value. 

§  o.  The  second  and  the  last  kind  of  value  Avere  distin- 
guished by  Turgot,  Avho  called  the  former  "  valeur  estimative  " 
and  the  latter  "  A^aleur  echangeable."  "     The  first  and  tlie  last 

^  Beccariasiiid  that  originally  "  value  "  meant  "  having  force,  liahitude,  ability 
to  fulfil  a  purpose,"  and  this  "  absolute  value  "  later  became  "  relative  and  venal  " 
and  meant  "  the  power  which  everything  has  of  being  exchanged  with  all  others," 
Del  commerc'io,  written  about  176!),  ed.  Custodi,  Vol.  I.,  p.  339.  These  two  senses 
are  not  specifically  distinct,  but  the  latter  is  one  species  under  the  former  as 
genus.    The  other  species  were  not  noticed. 

2  Valeurs  et  monnaies,  about  \7nQ,  ed.  Daire,  pp.  82,  S3.  lie  also  called  the 
later  "  valeur  appreciative,"  because  we  generally  estimate  the  exchange-value  of 
things  by  appraising,  or  setting  prices  on,  them,  p.  87. 


4  THE  NATURE  OF  EXCHANGE- VALUE 

were  distinguished  by  Adam  Smith,  who  called  the  first "  value 
in  use  "  and  the  last  "  value  in  exchange  "  or,  obviously  bor- 
rowing from  Turgot,  "  exchangeable  value."  ^  But  the  latter 
term  Adam  Smith  did  not  confine  to  the  meaning  of  exehang-e- 
value.  He  divided  "  value  in  exchange  "  into  two  sub-species, 
which  he  called  "  nominal  value  "  and  "  real  value  "  or  even 
"  real  exchangeable  value."  By  the  former  of  these  he  meant 
merely  one  instance  of  exchange-value,  namely  a  thing's  ex- 
change-value in  one  special  other  thing,  money, — the  reverse 
of  its  "  nominal  price,"  the  quantity  of  money  we  must  pay  to 
purchase  it ;  and  by  the  latter  he  meant  the  quantity  of  labor 
the  thing  will  purchase, — the  reverse  of  its  "  real  price,"  the 
quantity  of  labor  it  costs  to  acquire  the  thing.^  Thus  what  is 
here  called  esteem-value  was  vaguely,  and  by  a  means  of  meas- 
uring it,  rather  than  by  its  nature,  referred  to  as  ''  real  value  "; 
and  what  is  here  called  cost-value  was  given  an  entirely  differ- 
ent appellation,  "  real  price."  ^  While  Adam  Smith's  first  and 
incomplete  dichotomy  of  "  value  "  has  been  widely  accepted, 
his  distinction  between  "  real  value  "  and  "  real  price  "  has 
hardly  even  been  noticed.^  By  Ricardo  the  term  "  real  value  " 
was  applied  to  cost- value,  when  he  spoke  of  it  as  "  the  quantity 
of  labor  and  capital  " — itself,  in  his  view,  a  product  of  labor — 
"  employed  in  producing  "  the  thing,' — in  which  sense  he  also 
indifferently  employed  the  term  "  natural  price."  ^  This  kind 
of  value  he  never  differentiated  from  esteem-value,  although  all 

3  Wealth  of  nations,  1776,  McCuUoch's  ed.,  1858,  p.  13. 

*See  especially  pp.  13,  lC-17,  97,  157. 

*  Tiie  explanation  of  this  last  term  is  to  be  found  on  p.  14.  The  idea  of  it  was 
evidently  drawn  from  Turgot,  who  had  said  that  things  would  have  value  to  an 
isolated  man  in  so  far  as  they  cost  him  trouble  to  obtain  them,  and  had  spoken  of 
a  commerce  between  man  and  Nature,  who  exacts  from  him  labor  as  t\\ii  price  he 
must  j:)«2/  for  what  she  yields,  op.  cit.,  pp.  82-83.  This  mere  figure  of  speech  was 
taken  literally  by  Adam  Smith. 

^  Adam  Smith  always  used  "price"  in  the  sense  of  what  we  give  (or  would 
have  to  give)  in  exchange  for  what  we  acquire  (or  want),  and  "  value  "  in  the 
sense  of  what  we  wc^Jt/re  (or  could  acquire)  in  exchange  for  what  we  give  (or 
have).  This  indeed  is  not  a  useful  distinction  to  observe  between  these  terms, 
and  has  rightly  been  rejected.  Yet,  along  with  the  realism  about  labor  as  an  ex- 
changeable commodity,  it  is  at  the  bottom  of  many  of  Adam  Smith's  doctrines. 

'  Works,  McCulloch's  ed.,  p.  32;  cf.  also  p.  171. 

« Ibid.,  p.  49. 


FOUR    KINDS    OF    ECONOMIC    VALUE  5 

his  investigations  into  the  causes  of  variations  in  what  he  called 
"  relative  value  "  hit  upon  the  causes  of  variations  in  the  rela- 
tive esteem-values  of  things.  Thus  what  is  here  called  ex- 
change-value he  often  referred  to  by  the  term  "  relative  value  "; 
and  he  also  occasionally  spoke  of  it  as  "  nominal  value,"  namely 
as  value  "  in  coats,  hats,  money,  or  corn,"  /,  c,  in  various  ob- 
jects named. ^  Like  Adam  Smith,  he  does  not  expressly  tell 
us  that  he  is  subsuming  these  values  under  "  exchangeable 
value  ";  yet  he  too  must  do  so,  as  he  made  his  first  division  of 
"  value  "  likewise  only  into  "  value  in  use  "  and  "  value  in  ex- 
change," '^  and  it  is  apparent  they  cannot  be  subsumed  under 
"  value  in  use,"  which  he  identified  with  "  riches."  '^  More- 
over there  are  passages  in  which  he  uses  "  exchangeable  value  " 
as  he  elsewhere  uses  "  real  value,"  ^^  while  it  is  evident  that 
"  exchangeable  value  "  must  mean  "  relative  value  "  when  he 
defines  it  as  "  the  power  of  purchasing  [other  things]  pos- 
sessed by  any  one  commodity."  '^ 

This  error  of  confounding  at  times  two,  or  even  three,  dis- 
tinct kinds  of  value  together  under  a  term  applicable  only  to 
one  of  them,  along  with  that  of  assigning  principal  importance, 
implied  in  the  term  "  real,"  to  the  very  one  which  does  not  de- 
serve to  come  under  that  term,  has  run  through  almost  all  the 
so-called  "  classic  "  political  economy,  notwithstanding  that  this 
school  has  constantly  claimed  exchange- value  to  be  the  chief 
topic  it  dealt  with,  and  has  rarely  treated  of  "  value  "  except 

^  Ibid.,  p.  32.  "  Value  "  was  sometimes  used  by  Eicardo  also  in  the  sense  of 
what  is  in  this  work  called  general  exchange- value,  pp.  293,  401 ;  but  he  also  de- 
nied this  use,  p.  171,  and  ignored  it,  p.  13. — As  for  "  nominal  value,"  this  term  was 
best  defined  by  a  contemporary  economist :  "  The  nominal  value  of  a  commodity 
is  strictly  speaking  its  value  in  any  one  commodity  named  ;  but  as  the  precious 
metals  are  on  almost  all  occasions  the  commodity  named,  or  intended  to  be  nanjed, 
the  nominal  value  of  a  commodity,  when  no  object  is  specifically  referred  to,  is 
always  understood  to  mean  its  value  in  exchange  for  the  precious  metals,"  ^lal- 
thus,  Princij)Ies  of  political  economy,  2d  ed.,  1831),  p.  ."i4.  This  narrow  sense  is  the 
one  above  seen  to  have  been  used  by  Adam  Smith.  For  it  "  money-value"  is  a 
shorter  term,  and  free  from  all  ambiguity.  Even  in  its  wider  sense  the  term 
"  nominal  value  "  is  not  satisfactory. 

10  Works,  p.  P. 

'^'^  Ibid.,  pp.  1(59-173,  against  J.  B.  Say,  who  had  sometimes  identified  "  real 
value"  with  "value  in  use." 

^~  Ibid.,  pp.  172,  377. 

^^  Ibid.,  p.  49. 


6  THE    NATURE    OF    EXCHAXGE-VALUE 

under  the  heading  of  Exchange  or  Distribution,  and  never 
under  the  heading  of  Production,  which  woukl  seem  to  be  the 
proper  place  for  treating  of  cost-value,  nor  under  the  heading 
of  Consumption,  which  would  seem  to  be  the  proper  place  for 
treatina:  of  esteem-value.  Hence  it  has  been  the  cause  of  un- 
told  amount  of  confusion  of  thought,  and  of  wasted  effort  to  get 
straightened  out.  Recognition  of  the  distinction  between  all 
four  kinds,  and  of  the  at  least  coordinate  importance  of  the  last 
kind,  has  not  yet  come  into  general  consciousness,  although  a 
beginning  has  been  made.  It  was  not  till  a  little  over  twenty 
years  ago  that  three  of  the  four  kinds  were  distinguished. 
Then  Jevons  separated  from  each  other  use- value,  esteem-value, 
and  exchange-value,  under  the  titles  of  "  value  in  use,"  "  es- 
teem, or  urgency  of  desire,"  and  "  purchasing  power,  or  ratio 
of  exchange."  ^*  It  is  high  time  that  all  four  should  be  distin- 
guished, and  their  distinctness  observed. 

It  woidd  be  out  of  place  in  this  work  to  pursue  the  distinc- 
tion further.  Of  all  the  four  kinds,  the  last  is  the  only  one 
which  has  always  been  treated  of  by  economists,  and,  as  just 
observed  in  another  form,  many  of  them  have  asserted  that 
their  science  is  specially  and  even  wholly  concerned  with  ex- 
change-value. In  this  position  may  be  some  exaggeration, 
especially  when  we  sever  from  exchange-value  the  other  kinds 
of  value  which  they  unconsciously  associated  with  it.'^  It  is 
proper,  however,  that  special  treatises  should  be  confined  to  this 
one  particular  kind  of  value,  and  it  is  the  whole  and  sole  subject 
of  the  present  work. 

'"  Theory  of  political  economy,  2d  ed.,  1879,  pp.  85,  87,  3d  ed.,  1888,  pp.  78,  81. 
These  terms  are  not  in  the  1st  ed.  published  in  1871.  But  in  the  1st  edition  he  had 
really  made  the  same  distinctions  in  thought.  At  the  same  time  Walras  was  work- 
ing out  the  laws  of  esteem-value  in  very  much  the  same  way,  publishing  them  in 
the  1st  edition  of  his  ElemeMs  d'economie  politique  pure,  1874.  While  Jevons 
generally  used  the  term  "value"  confined  to  the  meaning  of  exchange-value, 
W'alras  has  employed  it  mostly  in  the  sense  of  esteem-value,  occasionally  using 
the  term  "  exchangeable  value  "  when  treating  of  exchange- value,  and  frequently 
employing  long  and  tedious  phrases,  from  lack  of  short  and  clear-cut  terms 
whereby  to  distinguish  the  kinds  of  value  he  had  in  mind,  and  sometimes  fall- 
ing into  confusion  in  consequence  of  dropping  the  long  phrases. 

^^  As  political  economy  does  deal  with  all  the  kinds  of  value,  a  better  technical 
name  for  it  than  ^Vhateley's  "catallactics"  would  be  "  timiotology,"  although 
perhaps  Hearn's  "  plutology  "  is  still  better. 


THE    RELATIVITY    OF    EXCHANGE-VALUE  7 

II. 

§  1.  Exchange- value  is  a  relative  quality  in  material  things. 
A  material  thing  has  exchange- value,  as  it  has  weight,  only  be- 
cause of  other  material  things  to  which  it  relates  in  a  particular 
way.  This  is  its  exchanging  for  them,  or  its  ability  to  exchange 
for  them.  Gravity  is  the  power  in  a  thing  by  which  it  attracts 
other  things  toward  itself  and  is  attracted  toward  other  things 
by  a  similar  power  in  them.  Exchange-value  is  the  power '  in 
a  thing  by  which  it  procures  for  its  owner  other  things,  which 
procure  it  for  their  owners  by  a  similar  power  in  them.  As  we 
cannot  conceive  of  the  gravity  of  one  thing  alone,  without  refer- 
ence to  other  things,  we  cannot  conceive  qf  the  exchange-value  of 
one  thing  alone,  without  reference  to  other  things.^  Further- 
more, in  the  case  of  exchange-value,  we  know  that  for  its  ex- 
istence is  required  something  else,  namely  the  men  who  make 
exchanges.^  But  it  is  likewise  believed  that  for  the  existence 
of  weight,  or  attraction,  in  material  things,  there  is  required 
some  other  thing  as  its  cause,  which  has  been  variously  placed. 
Now  just  as  we  can  conceive  of  weight  or  attraction  without 

^  Cf.  Beccaria  andRicardo  above.  ileCulloch  speaks  of  "exchangeable  value 
being  the  power  which  a  commodity  has  of  exchanging  for  other  commodities," 
Principles  of  political  ecoyiomy,  1825,  p.  213  (repeated  in  his  ed.  of  the  Wealth  of 
nations,  p.  439).  Courcelle  Seneuil :  "  The  value  of  a  commodity  is  the  force  or 
power  of  exchange  of  this  commoditj',"  Truife  d' economie  politique,  1858,  Vol.  I., 
p.  256.  And  Walras  speaks  of  a  relation. establishing  itself  between  appropriated 
things  such  that  "each  one  acquires,  as  a  special  property,  a  faculty  of  exchang- 
ing for  each  of  the  others  in  definite  proportions,"  Elements,  1st  ed.,  pp.  25-26; 
and  on  p.  48  he  applies  this  "  property  "  to  "  exchangeable  value." 

2  Yet  Bourguin  says  that  exchange-value  is  not  a  property,  a  quality,  an  at- 
tribute, of  things — there  is  no  intrinsic  value,  that  is,  something  which  we  can 
conceive  of  in  an  isolated  body  as  a  quality  inherent  in  it  independently  of  all 
other  things,  adding  "length  and  weight  can  be  so  conceived  in  a  body,  apart  from 
any  relation,  from  any  comparison  with  another  thing:  they  are  therefore  intrin- 
sic qualities,"  B.  132,  pp.  22-23 ;  cf.  p.  208.  This  is  curious.  An  isolated  body 
could  not  have  weight,  and  we  cannot  conceive  of  its  having  weight,  e.  g.,  being 
unsupported,  in  which  direction  would  it  foil?  As  for  length,  we  can  conceive  of 
an  isolated  body  having  length  provided  we  conceive  of  it  having  many  distinct 
parts,  themselves  not  isolated.  Length  is  properly  the  distance  between  such 
parts,  and  distance  is  not  an  intrinsic  quality  inherent  in  any  isolated  thing. 

'  The  existence  of  the  men  who  use,  prize,  and  produce  things,  is  also  necessary 
for  the  attaching  of  the  other  kinds  of  "  value  "  to  things.  But  any  material  thing 
can  have  use-value,  esteem-value,  or  cost-value,  without  reference  to  any  other 
material  thing. 


8  THE    NATURE    OF    EXCHANCJE-VALUE 

bringing  into  the  question  consideration  of  that  other  thing,  so 
wc  can  conceive  of  exchange- value,  or  of  exchanges,  without 
bringing  into  the  question  consideration  of  the  men  who  make 
them/  At  all  events  in  our  present  limited  incjiiiry  it  is  un- 
uecessarv  to  investigate  the  relationship  between  valnable  things 
and  their  owners,  or  the  motives  by  which  these  are  actuated  in 
exchanging  them  in  certain  quantities.  The  owners  are  agents, 
and  their  motives  are  determining  reasons,  for  the  making  of 
exchanges,  and  consequently  for  attaching  exchange-value  to 
things.  In  seeking  to  compare  and  measure  the  exchange- 
values  already  formed  and  put  into  things,  comformably  to 
qualities  already  under  given  conditions  found  existing  in  the 
things,  we  no  more  need  to  know  the  causes  of  exchange-value 
than  we  do  to  know  the  cause  of  gravitation.  Indeed  psychology 
holds  somewhat  the  same  relation  to  objective  or  formal  eco- 
nomics (the  study  of  the  phenomena  of  exchanges  and  the  laws 
of  their  relations)  as  theology  to  physics.  We  may  make  use 
of  psychology  in  some  branches  of  economics,  and  get  back  to 
causes  not  manifested  in  the  phenomena  themselves.  This 
means  that  we  can  go  further  in  economics  than  we  can  in 
physics.      But  for  merely  measuring  the  relations  of  exchange- 

•*  J.  B.  Clark  :  "The  inaccuracy  of  the  term  purchasing  power,  often  used  as 
synonyn^ous  with  value  in  excliange,  consists  mainly  in  its  implying  a  power  in 
the  commodity  itself  to  effect  a  purchase.  Such  power  resides  in  men,  not  in 
things,"  The  philosophy  of  wealth,  1886,  p.  88.  To  be  sure,  the  power  of  eft'ecting 
exchanges  (the  causa  agendi)  resides  only  in  men  ;  but  witliout  possessing  some 
material  thing  men  have  no  power  to  purchase  anything  (although  they  may  have 
power  to  produce  or  to  earn  something).  i\Iore  fully  described,  exchange-value 
is  the  power  in  things  to  be  taken  in  exchange  for  other  things  (the  causa  fiemli). 
It  is  a  power  in  tilings  by  means  of  which  their  possessors  have  power  of  effecting 
an  exchange.  A  derrick  has  no  power  "to  effect  the  lifting  "  of  anything ;  yet  it 
has  a  power  by  means  of  which  men  can  effect  tiie  lifting  of  things.  It  would 
seem  as  if  F.  A.  Walker  tried  to  avoid  Clark's  objection  wlien  he  defined  "  value  " 
as  "  the  power  which  an  article  confers  upon  its  possessor  .  .  .  of  commanding, 
in  exchange  for  itself,"  other  things.  Political  economy,  1887,  pp.  5,  81.  This 
would  place  value  in  the  possessor,  which  is  absurd.  The  same  idea  is  expn^ssed 
l)y  Bourguin,  who,  however,  immediately  ajjpended  the  old  opinion  tiiat  "  value  " 
resides  in  the  commodity,  saying  that  it  is  "the  commodity's  excliangeability  for 
anotlicr  commodity,"  B.  132,  p.  20.  It  is  interesting  to  note  that,  without  think- 
ing of  tiiis  question,  Adam  Smith  described  "  value  in  exchange  "  as  "  the  power 
of  purchasing  other  goods,  which  tlie  possession  of  that  object  [one  liaving  utility] 
conveys,"  op.  cit.,  p.  18.  This  means  tliat  neither  the  object  nor  the  possessor,  but 
the  possession  has  exchange-value.  To  conform  t"  this  position  in  our  speech 
would  only  be  to  use  much  pleonasm. 


THE    KEI.ATIYITY    OF    f:X(irANGE-VALUE  9 

value,  as  manifested  in  actual  exchanges,  we  might  be  disem- 
bodied .spirits  investigating  tiie  laws  of  a  world  in  which  we  had 
no  part  and  in  which  we  could  not  go  behind  the  scenes. 

§  2.  The  assertion  that  exchange-value  is  a  quality  of  mate- 
rial things,  or  in  material  things,  is  often  denied  on  the  ground 
that  exchange-value  is  only  an  estimation  which  men  set  upon 
things  and  so  is  only  in  our  minds — only  subjective,  not  at  all 
objective.  To  maintain  this  denial  is  merely  to  employ  meta- 
physics in  the  wrong  place.  For  on  this  line  of  reasoning 
there  are  no  qualities  in  things,  since  there  is  no  quality  said  to 
be  in  things  known  to  us  but  it  requires  our  presence  for  its 
existence  as  we  know  it.  We  are,  however,  permitted  by  meta- 
physics to  speak  of  qualities  of  or  'ni  material  things,  on  the 
ground  that  the  material  things  themselves  are  in  us  in  the 
same  sense  in  which  it  is  said  the  qualities  we  assign  to  them 
are  in  us.  We  may,  then,  be  permitted  to  continue  using  the 
same  popular  phraseology  in  all  cases,  and  to  speak  of  the  ex- 
change-value (or  other  values)  of  things  so  long  as  we  speak  of 
the  weight,  size,  hardness,  color,  etc.,  of  things.  It  is  perfectly 
correct  to  say  that  exchange-value  is  something  in  our  minds. 
But  it  is  also  correct,  as  we  all  really  believe,  to  say  that  ex- 
change-value is  something  in  things,  and  it  is  not  correct  to  say 
that  exchange-value  is  only  in  our  minds.  Exchange-value  is 
not  merely  subjective ;  it  is  also  objective.  We  believe  there 
is  something  in  material  things  (whether  these  be  in  us  or  not) 
that,  along  with  our  own  constitution,  is  the,  or  a,  cause  of  our 
desiring  them  and  behaving  toward  them  as  we  do  ;  and  it  is 
this  somethino;  in  them  to  which  we  refer  when  we  think  of 
their  value,  or,  in  particular,  of  their  exchange-value.'' 

^  H.  D.  Macleod  :  "  Value  is  not  a  quality  of  an  object  ...  it  is  an  affection 
of  the  mind.  The  sole  origin,  form,  or  cause  of  value  is  human  desire,"  Theory 
of  credit,  1893,  Vol.  I.,  p.  200.  This  omits  to  say  what  is  the  cause  of  our  desire 
for  things.  Surely  our  desire  for  things  is  not  wholly  independent  of  the  things 
themselves.  W.  L.  Trenholm :  "It  is  obvious  that  it  cannot  be  any  special 
quality  in  the  thing  desired  which  gives  value  to  it,  but  that  the  value  comes 
wholly  from  unsatisfied  desire,"  The  2)eople's  money,  1S93,  Tp.  226.  This  ignores 
the  connection  between  the  "special  quality  "  and  the  desire. — C.  Menger  would 
also  deny  the  existencre  of  exchange-value  in  things,  affirming  that  it  is  not  a  "  real 
phenomenon,"  not  only  because  it  cannot  exist  in  an  isolated  body,  but  liecause 
it  is  "  the  thought-of  cause  of  the  existing  various  exchange  relations  and  of  their 


10  THE    XATTTRE    OF    EXCHANGE- VALUE 

This  whole  question,  liowcver,  thouoh  important  in  meta- 
physics, is  of  no  consequence  in  economics  f  for  we  should  con- 
tinue to  think  of,  and  to  investigate  the  relations  of,  exchange- 
value  in  precisely  the  same  manner  whether  metaphysicians 
decided  there  is  exchange-value  in  things  or  whether  they  de- 
cided it  is  only  in  our  minds.  The  difficulties  which  confront  us 
in  the  metrology  of  exchange- vahie  do  not  arise  from  its  sub- 
jectivity, if  it  be  only  subjective,  and  would  exist  the  same  if  it 
be  also  objective.  They  arise  solely  from  its  extreme  variabil- 
ity, whatever  be  the  cause  of  this.  But  it  has  been  necessary 
to  point  this  out  here,  on  account  of  the  frequency  with  which 
the  opinion  is  advanced  that  we  can  find  no  invariable  standard 
of  exchange-value  because  of  its  being  only  subjective  unlike 
weight  and  other  qualities  which  we  have  succeeded  in  measur- 
ing with  tolerable  exactness." 

§  3.  A  thing  which  has  exchange-value  has  exchange-value 
in  relation  to  other  particular  things,  or,  by  combining  these 
under  the  same  terms,  to  other  particular  kinds  of  things. 
Individuals  of  one  kind  exchanging  in  certain  proportions  for 
individuals  of  this  or  of  that  or  of  any  other  kind,  it  is  said 
the  former  kind  has  a  certain  exchange-value  in  this  kind,  in 

variations,  which  cause  is  in  our  thoughts  ascribed  to  the  commodity  in  ques- 
tion," art.  Geld  in  the  Handworterbuch  der  Staatswissenscliaften,  .Jena,  Vol. 
III.,  1892,  p.  740.  But  at  the  same  time  he  allows  it  to  be  an  Aiistauschm'nglich- 
i-e)7  l)elonging  to  things.  Tlien  why  not  also  a  Tauschkraft  F  And  if  we  define 
exchange-value  as  such,  it  would  belong  to  things. — Tlie  denial  is  made  also  on 
the  ground  that  exchange-value  is  only  a  relation.  8o  Denis,  B.  100,  p.  171. 
The  error  of  this  will  be  seen  presently. 

'^  Of  course  to  complain,  with  H.  Dabos,  La  theorie  de  la  valeicr,  .Journal  des 
Economistes,  Marcli,  1888,  p.  406,  that  exchange-value  is  not  a  pliysical  property 
of  tilings,  like  their  color,  density,  porosity,  etc.,  is  to  go  out  of  one's  way  in 
search  of  trouble,  since  economists  should  leave  physical  properties  to  physicists 
and  concern  themselves  with  the  economic  properties  of  things. 

'  E.  g.,  .J.  P.  Smith,  B.  7,  p.  39.— T.  Jilartello  says  that  speaking  of  a  fixed 
unit  of  value  is  like  speaking  of  a  fixed  unit  of  love,  La  moneta  e  gli  errori  che 
corroHo  intorno  ad  essn,  1881,  p.  401. — It  is  only  a  step  to  say  tliat,  because  value 
is  only  in  us,  there  is  no  such  tiling  as  the  value  of  a  material  object.  This  step 
lias  been  taken  even  by  Walras  in  the  very  work  in  which  he  started  out  by  de- 
scribing exchange-value  as  a  property  acquired  by  things  under  certain  circum- 
stances; whence  he  concluded  that  the  term  "  franc"  cannot  refer  to  the  value 
of  a  piece  of  metal,  there  being  none,  but  only  to  the  piece  of  metal  itself, 
Elements,  p.  147  (2d  ed.,  p.  172).  But  this  has  not  prevented  him  from  later 
writing  a  work  on  a  method  of  regulating  variations  in  tiie  "  value  of  money," 
B.  69.     A  similar  position  is  adopted  by  Bourguin,  B.  132,  p.  38. 


THE    RELATIVITY    OF    EXC'IIAXGE-VALUE  11 

that  kind,  and  so  on, — a  contracted  expression  for  "  in  relation  to 
the  other  kind."  The  excliange-vahie  of  one  thing  in  another, 
or  of  one  kind  of  thing  in  another  kind  of  thing,  may  be  called 
a  particular  exchange-value  o^  the  thing,  or  of  the  kind  of  thing  ; 
so  that  we  may  speak  either  of  the  particular  exchange-value 
of  one  thing  in  another  thing  or  of  the  particular  exchange- 
value  of  one  kind  of  thing  in  another  kind  of  thing.* 

Such  a  particular  exchange- value  of  one  thing  in  another,  or 
of  one  kind  in  another,  is  not  the  other  thing  itself,  or  the 
definite  quantity  of  the  other  kind  of  thing,  procurable  in  ex- 
change, as  has  often  been  carelessly  said.^  Xor  is  it  simply  a 
relation,  or  ratio,  between  the  two  things.'"  Indeed  it  is  diffi- 
cult to  see  what  meaning  there  is  in  such  a  definition  of  ex- 
change-value. If  a  bushel  of  wheat  exchanges  for  two  bushels 
of  barley,  is  the  exchange-value  of  wheat  the  ratio  two  f  We 
commonly  say  in  this  case  that  the  exchange-value  of  wheat  is 
twice  that  of  barley  ;  but  this  is  only  a  statement  of  comparison 
between  the  exchange- values  of  equal  quantities  of  wheat  and 
of  barley.  It  does  not  pretend  to  tell  us  what  the  nature  is  of 
the  exchange-value  of  either  of  these  things."     It  is  ])lain  that, 

*  This  is  another  term  for  "  nominal  value  "  as  above  defined  V)y  ^lalthus. 

'  E.  g.,  Adam  Smith,  as  already  observed  in  Xote  6  in  Sect.  I.;  also  J.  B.  Say, 
Traite  d'  economie  jiolitique,  5th  ed.,  1826,  Vol.  II.,  p.  156  ;  .1.  S.  Mill,  Principles 
of  political  economy,  1848,  ed.  1878,  Vol.  I.,  p.  588.  Macleod,  op.  cit..  Vol.  I., 
pp.  112-113,  170,  172. — The  eri-or  has  been  pointed  out  by  Jevons,  oj).  cit.,  l.st  ed., 
pp.  81-82,  3d  ed.,  p.  78. 

'"  JE".  g.,  "Value  in  exchange  is  the  relation  of  one  object  to  some  other  or 
others  in  exchange,"  Malthus,  op.  cit.,  p.  50,  and  similarly  again,  p.  Gl. — Beccaria 
had  said:  "Value  indicates  the  proportion  of  one  quantity  with  another,"  op. 
cit.,  Vol.  II.,  p.  8.    There  is  a  great  difference  between  these  expressions. 

11  Thus  Jevons  himself  was  in  error  when  he  added  :  "  Value  in  exchange  is 
nothing  but  a  ratio,  and  the  term  should  not  be  used  in  any  other  sense,"  loc.  cit. 
All  that  he  had  a  right  to  say  was  that  the  ideas  of  the  other  kinds  of  value  should 
be  excluded  from  the  idea  of  this  kind  of  value.  He  had  himself  previously  de- 
clared of  this  kind  of  value:  "Value  is  a  vague  exjjression  for  potency  in  purchasing 
other  commodities,"  B.  22,  p.  20  ;  and  he  continued  to  use  the  term  in  this  sense, 
cf.  Investigations,  p.  358.  What  absurdity  this  position  may  lead  to  is  well  illus- 
trated in  the  following:  "Value  ...  is  a  relation  between  certain  things  to 
which  men  attribute  value,"  Trenholm,  op.  cit.,  p.  246. — Walras  has  gone  to  a 
peculiar  extreme  also  here.  He  says:  "  Strictly  speaking,  there  are  no  values, 
there  are  only  relations  of  values,"  Elements,  1st  ed.,  p.  180  (again  in  B.  71,  p.  4). 
(Perhaps  the  meaning  intended  to  be  conveyed  in  this  sentence  is  that  exchange- 
value  is  a  relation  between  esteem-values.) — Carelessness  in  trebly  describing 
value  as  (1)  a.  ratio,  (2)  the  quantity  of  another  or  other  things,  (3)  purchasing 


12  THE    NATURE    OF    EXCHANGE-VALUE 

as  already  shown  of  exchange-value  in  general,  the  exchange- 
value  of  one  thing  in  another  is  the  power  in  the  one  thing  of 
exchanging  for  the  other — a  power  which  necessarily  presup- 
poses certain  relations  of  (juantity  between  the  things  exchanged, 
but  is  itself  very  different  from  those  relations.  Thus  the  ex- 
change-value of  one  thing  in  another  is  neither  the  other  thing 
nor  the  relation  between  the  two  ;  but  it  is  the  power  in  the  one 
which  can  exist  only  in  connection  with  a  similar  power  in  the 
other,  and  can  be  estimated,  as  we  shall  see,  by  the  relation  be- 
tween the  quantities  exchanged. 

§  4.  A  thing  has  many  such  particular  exchange-values — as 
many,  in  fact,  as  there  are  kinds  of  things  with  which  it 
can  exchange.  Now  several,  many,  or  all  of  these  particular 
exchange- values,  as  they  exist  together,  may  be  combined  into 
a  single  concept,  and  so  provide  us  with  the  idea  of  the  thing's 
exchange- value  simply  so  called,'"  This  idea  of  simple  ex- 
change-value, like  that  of  the  gravity  of  the  heavenly  bodies,  is 
difficult  to  grasp  at  first.  Yet  it  is  a  necessary  idea,  which  we 
all  do  inevitably  form  with  various  degrees  of  definiteness  and 
accuracy.'^     It  is  at  any  given  time  and  place  a  single  exchauge- 

power,  is  also  shown  by  J.  L.  Laughlin,  Facts  about  money,  Chicago,  1895,  pp.  75- 
76,  147,  192,  and  Parsons,  B.  136,  pp.  81-82. 

*-  .T.  L.  Shadwell :  "  The  human  mind  can  only  compare  two  things  at  once, 
and  when  it  is  said  that  a  commodity  has  a  certain  power  of  purchasing  all  other 
commodities,  the  words,  though  they  may  be  pronounced,  written,  and  printed, 
do  not  really  present  any  idea  to  the  mind.  The  power  of  gold  to  purchase  silver 
is  a  definite  idea,  and  so  is  its  power  to  purchase  copper  ;  but  the  power  of  gold 
to  purchase  silver  and  copper  means  nothing  at  all,"  System  of  political  economy, 
London,  1877,  p.  93.  The  only  reason  offered  why  we  cannot  strike  an  average  is 
that  "  we  have  no  standard  by  which  to  measure  the  objective  importance  of  dif- 
ferent articles,"  p.  95.  This  is  a  difficulty,  to  be  sure,  but  one  by  no  means  in- 
supcra])le, — -and  one  wliich  has  long  been  discussed,  it  forming  one  of  the  definite 
problems  in  our  subject  (hei'e  to  be  treated  in  Chapter  IV.).  Few  persons  would 
admit  in  their  own  case  the  impotency  here  claimed  for  every  human  mind. 

^^  Bourguin  asserts  that  a  thing  has  no  "  valuein  general,"  butonly  "  particular 
values"  (by  "  value  "  always  meaning  "  exchange-value,"  B.  132,  p.  3).  His  only 
reason  seems  to  be,  because  exchange-value  is  not  a  property  inherent  in  things, 
p.  22.  Cf.  p.  135,  where  he  says:  "The  [purchasing]  power  of  money  is  only  a 
word ;  it  designates,  not  a  quality,  an  intrinsic  value,  but  an  ensemble  of  rela- 
tions which  have  nothing  in  common,  not  being  equations  between  magnitudes  of 
the  same  kind."  But  if  the  particular  exchange-values  are  not  intrinsic,  neither 
would  the  gi'ueral  exchange-value  be  intrinsic,  and  no  reason  is  shown  why  there 
is  no  non-intrinsic  general  exchange-value  except  the  statement  that  the  relations 
(i.  e.,  the  particular  exchange-values)  have  nothing  in  common.    But  this  is  not  so, 


THE    EET.ATIYITY    OF    EXCHANGE-VALUE  13 

value  somehow  made  up  of  many  particular  exchange-values. 
A  thing  has  many  exchange-values  in  other  things  separately  ; 
it  has  one  exchange-value  in  other  things  collectively.  Or 
rather  the  more  correct  statement  is  tliat  when  we  do  reach  this 
idea  of  a  thing's  simple  exchange-value  we  must  view  it  as  the 
thing's  only  real  exchange-value/*  and  regard  the  particular 
exchange-values  as  merely  this  one  and  the  same  exchange- 
value  in  its  various  relations  to  the  similarly  single  and  simple 
exchange- values  of  other  particular  things.  One  of  the  tasks 
of  political  economy  is  to  explicate  and  render  more  intelligible 
this  conception  of  a  thing's  exchange-value  in  other  things,  or 
in  all  other  things.  To  contribute  to  the  accomplishment  of 
this  task  is  one  of  the  objects  of  these  pages.  It  may  be  pre- 
mised that  we  shall  find  an  exchange-value  of  a  thing  in  all 
things,  including  itself, — an  idea  never  yet  distinguished  from 
that  of  the  exchange-value  of  a  thing  in  all  other  things.  These 
two  ideas  have  some  points  of  contact,  and  both  may  lay  claim 
to  the  title  of  a  thing's  exchange-value  simply,  or  its  general  ex- 
change-value.^^ It  is  plain  that  we  have  no  right  to  speak 
simply  of  a  thing's  exchange-value,  if  we  have  in  mind  only  its 
exchange-value  in  some  other  thing,  or  in  a  few  other  things. 
Speaking  of  its  exchange-value  simply,  we  should  be  using 
language  wrongly  unless  we  referred  to  its  exchange- value  in 
all,  or  in  all  other  things — or  at  least  in  all  others  to  which  we 
practically  can  refer, — that  is,  to  its  one  simple  exchange-value 
as  measured  by  comparison  with  the  simple  exchange-values  of 
all  the  other  things.'" 

as  the  particular  cxcliange-values  of  a  thing  have  in  common  powers  of  exclianging 
for  certain  quantities  of  otlier  tilings,  and  these  powers  are  magnitudes  of  the  same 
kind.  As  well  say  the  attraction  of  the  earth  for  the  moon  is  different  in  kind 
from  the  atti'action  of  the  earth  for  the  sun.  Bourguin  elsewhere  says  he  will  use 
the  term  "the  value  of  a  thing,"  in  the  singular,  meaning  the  ensemhle  of  all  its 
particular  exchange-values,  only  for  convenience,  p.  23  (and  he  will  attempt  to 
measure  variations  in  t\\\s  ensemble,  pp.  138-139).  But  we  may  be  sure  that  when 
a  term  is  found  to  be  convenient,  it  expresses  an  idea  or  concept. 

I''  But  of  course  not  its  only  "  real  value." 

^^  This  term  was  much  used  by  .J.  S.  W\\\.  In  the  same  sense  "  general  value  " 
was  used  by  Ilallam,  View  of  the  state  of  Europe  in  the  Middle  Ages,  181G,  Chapt. 
IX.,  Part  II. 

'^  Naturally  in  speaking  of  exchange-value  simply  we  do  not  mean  something 
without  relation  to  any  other  things.    Neglect  of  this  has  led  Macleod  into  curious 


14  THE  nature:  op^  exchange-value 

III. 

§  1.  Exchange- value  is  quantitative.  A  certain  quantity  of 
one  thing  exchanges  for  a  certain  quantity  of  another  thing.  If 
it  exchanges  for  more  of  the  other,  its  exchange-value  in  the 
other  is  proportionally  greater.  If  it  exchanges  for  less  of  the 
other,  its  exchange-value  in  the  other  is  proportionally  smaller. 
If  it  exchanges  for  the  same  quantity  of  the  other,  its  exchange- 
value  in  the  other  is  the  same.  Thus  the  exchange-value  of 
one  thing  in  another  may  have  different  degrees  of  intensity. 
This  being  so  of  its  exchange- value  in  any  one  other,  it  is  so  of 
its  exchange-value  in  every  other,  and  consequently  of  its  ex- 
change-value in  all  other  things,  or  of  its  general  exchange- 
value.     And  so  with  the  exchange-values  of  everything. 

Such  variations  in  exchanges,  and  consequently  in  exchange- 
values,  be  it  said  in  passing,  may  occur  at  different  times  and 
places.  Consequently  to  speak  of  a  thing's  exchange-value  in 
another  thing,  or  in  general,  is  to  refer  to  its  exchange-value  at 
a  given  time  and  place.  Exchange- value  is  something  temporal 
and  local.  '  In  omitting  explicit  declarations,  we  generally  im- 
ply that  we  are  dealing  with  exchange-value  at  the  same  place 
and  are  referring  to  changes  happening  in  time. 

§  2.  From  the  above  comparisons  it  results  that,  flic  quantity 
of  one  thing  being  assumed,  its  exchange-value  in  another  is  pro- 
portional  to  the  quantity  of  tJie  other  it  exchanges  for  (Proposition 
I.).  Consequently  we  can  measure  the  exchange- value  of  one 
thing  in  another  at  two  dates  (or  places)  by  the  relative  quan- 
tities of  the  other  it  exchanges  for  at  the  two  dates  (or  places). 

error.  He  says :  AVluit  is  wanted  by  economists  who  seek  an  invariable  standard 
of  value  is  "  something  by  which  they  ean  at  once  decide  whether  gold  is  of  more 
value  in  A.  D.  30,  in  A.  D.  1588,  or  in  A.  D.  189:5;  in  Italy,  in  England,  or  in 
China;  without  reference  to  anything  else,"  Theory  of  credit.  Vol.  I.,  p.  212. 
And  so  he  had  long  before  said  :  "  As  no  single  body  can  be  a  standard  of  distance 
or  equality  [without  reference  to  others],  so  no  single  object  can  possil)ly  be  a 
standard  of  value"  without  reference  to  others,  Theory  and  practice  of  haiithig, 
1875,  Vol.  I.,  p.  16,  and  similarly  again,  p.  77.  Of  course  the  exchange-value  of 
a  single  body  with  reference  to  others  may  be  a  standard  of  exchange-value,  just 
as  the  length  of  a  single  body — a  certain  distance  between  its  extrenjities — com- 
pared with  otliers,  may  be  a  standard  of  length.  (But  some  of  the  economists 
referred  to  treated  of  cost- value,  and  so  took  account  only  of  cost  of  production.) 


THE    QUAXTITATTVENES8    OF    EXCHAXGE-VALUE  15 

These  quantities  of  the  other,  we  must  remember,  do  not  con- 
stitute the  exchange-value  of  the  one  in  that  other.  The  ex- 
change-value of  the  one  in  the  other  is  its  power  of  acquiring 
that  other  for  its  owner.  The  magnitude  of  this  power  over 
the  other  is  manifested  by  the  effect  it  accomplishes  in  exchang- 
ing for  that  other,  that  is,  by  the  quantity  of  that  other  thing  it 
acquires.^  Here  we  are  treating  merely  of  the  power  of  the  one 
thing  over  the  other,  or  its  exchange-value  in  the  other,  and 
therefore  need  pay  no  attention  to  the  power  of  the  other,  or  to 
its  exchange-value.  But  if  we  were  comparing  the  exchange- 
value  of  the  one  with  the  exchange-value  of  the  other,  or  trying 
to  measure  how  much  exchange-value  the  former  is  manifesting 
when  it  is  exchanging  for  the  latter — by  "  exchange-value"  here 
meaning  exchange-value  simply, or  general  exchange-value, — we 
should  have  to  take  into  consideration  also  the  general  exchange- 
value  of  the  latter. 

It  is  plain  that  at  any  one  place  individual  things  exactly  alike 
pJiysically  will  always  exchange  for  one  another  indifferently,  or 
have  the  same  exchange-value  in  their  own  class ;  and  also  will 
exchange  for  the  same  quantities  of  other  things,  or  have  the  same 
exchange-value  in  other  things :  that  is,  all  the  individuals  in  a 
homogeneous  class  have  the  same  exchange-value  (Proposition  II.). 
Like  things  are  not  generally  exchanged,  because  there  is  gen- 
erally no  object  in  making  such  an  exchange.  But  such  an  ex- 
change is  possible,  and  sometimes  occurs.  Dealers  who  store 
wheat  together,  probably  get  back  different  wheat,  and  so  have 
exchanged  wheat  for  wheat.  Also  in  the  case  of  money,  we  fre- 
quently exchange,  say,  ten  dollars  in  one  piece  for  ten  dollars 
in  two  or  more  pieces.  It  is  evident  that  when  such  exchanges 
are  made  people  do  not  exchange  more  for  less  of  the  same 
thing.  Now  a  bushel  of  wheat  and  any  other  bushel  of  wheat 
having  the  power  of  exchanging  for  a  bushel  of  wheat,  every 
bushel  of  wheat  has  the  same  exchange- value  in  wheat  (all  of 
the  same  quality).  And  when  one  kind  of  things  is  exchanged 
for  another  kind,  it  is  indifferent  which  of  many  like  individuals 
is  given  or  received.     Therefore,  each  of  these  in  the  one  class 

iCf.  Courcelle  Seneuil,  op.  cit..  Vol.  I.,  p.  243. 


16  THE  NATURE  OF  EXCHANGE- VALUE 

having  the  power  of  exchanging  for  the  same  thing  in  the  other, 
they  all  have  the  same  exchange-valne  in  that  other  kind  of 
thing.  And  for  the  same  reason  they  have  the  same  exchange- 
value  in  any  and  every  other  kind  of  things,  consequently  in 
all  other  kinds  of  things  together,  that  is,  in  exchange-value 
simply,  or  in  general  exchange-value.  It  may  be  that  the  dif- 
ferent individuals  Avhich  we  include  under  a  class  with  the  same 
name  are  generally  not  exactly  alike,  and  perhaps  they  never 
are  exactly  alike — it  has  been  maintained  that  no  two  grains  of 
wheat  are  ever  exactly  alike  ; — but  they  are  often  nearly  enough 
alike  to  pass  in  practice  for  alike.  Of  course  what  is  here  said 
refers  to  materials  in  the  same  form  only  ;  for  the  form  itself  has 
utility.  A  cubic  foot  of  wood  in  a  lumber  yard  has  not  the  same 
exchange-value  as  a  similar  cubic  foot  of  wood  in  a  building. 

It  is  plain  also  that,  the  exchange-value  of  a  guanfiti/  of  one 
thing  in  another  being  given,  the  power  of  acquiring  quantifies  of 
the  latter  by  means  of  the  former  is  proportional  to  the  quantity  of 
the  former  employed  in  exchanging  for  the  latter  (Proposition 
III.).  For  example,  if  one  bushel  of  wheat  has  the  power  of 
exchanging  for  two  bushels  of  barley,  two  bushels  of  wheat 
have  double  this  power,  that  is,  they  have  the  power  of  ex- 
changing for  four  bushels  of  barley  ;  for,  according  to  the  pre- 
ceding proposition,  the  second  bushel  of  wheat  has  the  same 
power  as  the  first.  This  proposition  does  not  mean  that  if  ten 
bushels  of  wheat  have  the  power  of  exchanging  for  a  diamond 
of  a  certain  size,  twenty  bushels  of  wheat  have  the  power  of 
exchanging  for  a  diamond  of  twice  that  size.  It  means  only 
that  they  have  the  power  of  exchanging  for  two  such  diamonds- 
It  also  does  not  mean  that  if  one  bushel  of  wheat  actually  ex- 
changes for  two  bushels  of  barley,  any  (luantity  of  wheat  might 
have  been  exchanged  for  twice  the  same  quantity  of  barley  ; 
for  this  would  require  that  the  exchange-values  should  remain 
the  same  whatever  be  the  quantities  offered  in  the  market.  It 
means  only  that  such  proportional  exchanges  can  be  made  while 
the  exchange-values  do  remain  the  same. 

§  3.  Again,  of  two  kinds  of  thixgx  the  qufoifitirs  vJiicJi  ex- 
change for  each  other  are  equivalent  in  the  .sense  find  llw  exchange- 


THE    QIWXTITATIVENESS    OF    EXTHANGE- VALUE  17 

value  of  the  one  in  the  otJier  is  equal  to  the  exchange-value 
of  the  latter  in  the  former  (Proposition  IV.).  This  looks 
as  if  it  were  an  expletive  proposition,  like  those  which  have 
preceded,  springing  from  the  meaning  of  the  terras  employed, 
or  like  saying  that  the  distance  of  the  snn  from  the  earth  is 
eqnal  to  the  distance  of  the  earth  from  the  sun.  Yet  it  may 
be  objected  that,  the  power  of  wheat  to  acquire  barley  being 
measured  by  the  quantity  of  the  barley,  and  the  power  of 
barley  to  acquire  wheat  being  measured  by  the  quantity  of  the 
wheat,  as  the  quantity  of  barley  and  the  quantity  of  wheat  are 
two  distinct  and  generally  unequal  things,  the  equality  of  the 
two  exchange-values  is  not  shown  by  a  comparison  of  these 
quantities.  If  a  proof  be  demanded,  however,  a  proof  is  forth- 
coming. AYe  have  seen  that  a  bushel  of  wheat  has  the  power 
of  acquiring  a  bushel  of  wheat.  But,  according  to  our  suppo- 
sition, two  bushels  of  barley  have  the  power  of  acquiring  a 
bushel  of  wheat.  Therefore  two  bushels  of  barley  have  the 
same  exchange-value  in  wheat  as  one  bushel  of  wheat.  And 
similarly  a  bushel  of  wheat  has  the  same  exchange-value  in 
barley  as  two  bushels  of  barley.  Therefore,  having  the  same 
exchange- value  both  in  barley  and  in  wheat,  a  bushel  of  wheat 
has  the  same  exchange-value  in  barley  as  the  two  bushels  of 
barley  have  in  wheat.- 

To  say  that  the  quj^tities  of  things  which  exchange  for  each 
other  are  equivalent  in  the  sense  of  having  the  same  exchange- 
value  simply,  that  is,  the  same  general  exchange-value,  needs 
still  further  proof. 

Now,  of  two  kinds  of  things  the  quantities  which  exchange  for 
each  other  exchange  for  the  same  quaniiiy  of  any  other  kind  of 
things,  and  therefore  have  the  same  exchange-value  in  that  other 
kind  of  things  (Proposition  V.).  The  first  part  of  this  proposi- 
tion is  the  law  of  stable  equilibrium  in  an  open  and  free  market, 
which  equilibrium  must  exist  on  the  average  in  the  long  run. 
It  is  possible  for  fluctuations  to  occur  at  times,  but  then  forces 
are  set  at  work  to  restore  the  equilibrium.     For  instance,  while 

^  The  cas6  is  comparable  with  weights.     "We  should  never  know  by  the  balanc- 
ing of  two  bodies  in  opposite  scales  that  they  are  of  equal  weight  except  by 
alternating  them,  or  by  employing  "  double  weighing." 
2 


18  THE    NATURE    OF    EXCHANGE-VAT,UE 

one  bushel  of  wheat  exchanges  for  two  bushels  of  barley  and 
for  three  bushels  of  oats,  if  it  should  happen  that  some  one  is 
willing  to  give  up  four  bushels  of  oats  for  two  bushels  of  bar- 
ley, immediately  those  who  have  wheat  and  want  oats  Avould 
exchange  their  wheat  for  barley  and  it  for  this  man's  oats,  and 
those  who  have  barley  and  want  wheat  would  exchange  their 
barley  for  this  man's  oats  and  them  for  wheat.  This  man's  sup- 
ply of  oats  would  then  be  soon  exhausted,  and  he  would  retire 
from  the  market.  The  condition  which  renders  useless  such 
roundabout  exchanges  (which  in  money  exchanges  between  dif- 
ferent places  are  called  arbitrages) — a  condition  which  those 
roundabout  exchanges  themselves  tend  to  produce, — exists  in  our 
example  when  three  bushels  of  oats  exchange  for  two  bushels 
of  barley.  Then,  representing  one  bushel  of  each  by  A,  B  and 
C,  and  equivalence  by  the  sign  --,  we  have  between  the  kinds 


2B 


lA 


3  C 


of  things  the  interrelation  here  depicted.  That  any  two  of 
these  quantities  have  the  same  exchange-value  in  the  third  is 
apparent. 

Tliis  being  so  of  any  two  things  in  cm j|  third,  it  must  be  true 
that  the  quantities  of  everything  which  exchange  for  each  other 
have  the  same  exchange-value  in  all  things  beside  themselves. 
But  as  their  own  exchange- values  are  equal  each  in  the  other, 
these  may  be  added,  and  we  have  :  Of  everythlmj  tlic  quantities 
wJiich  exchamje  for  each  other  have  the  same  exchange-value  in  all 
other  things  and  in  all  things,  that  is,  the  same  general  exchange- 
value,  or  the  same  exchange-value  simply  (Proposition  A^I.) ;  which 
is  what  we  wished  to  prove. 

It  follows  also  that  all  the  many  particular  exchange-values  of 
one  thing  in  other  things  singly  are  singly  equal  to  the  thing's  gen- 
eral exchange-value,  and  to  one  another  (Proposition  VII.).  The 
exchange-value  of  wheat  in  barley  is,  for  instance,  its  power  of 
acquiring  two  bushels  of  barley,  and  its  exchange-value  in  oats, 


THE    QUANTITATIVENESS    OF    EXCHANGE- VALUE  19 

its  power  of  acquiring  three  bushels  of  oats  ;  but  the  two  bushels 
of  barley  aud  the  three  bushels  of  oats  have  the  same  general  ex- 
change-value, therefore  the  one  bushel  of  wheat  is  acquiring  the 
same  exchange-value,  to  which  its  own  is  equal,  whether  it  be 
exchanged  for  two  bushels  of  barley  or  for  three  bushels  of  oats. 
It  is  evident,  moreover,  that  when  a  bushel  of  wheat  is  used  to 
acquire  barley,  it  is  manifesting  its  whole  exchange-value,  but 
only  in  relation  to  barley  ;  and  when  it  is  exchanged  for  oats,  it 
is  manifesting  its  whole  exchange-value,  but  only  in  relation  to 
oats.  Therefore  it  is  manifesting  the  same  exchange- value  in 
both  cases,  but  in  different  relations.  The  two  particular  ex- 
change-values, as  we  have  already  noticed,  are  the  same  as  the 
general  exchange- value,  are  equal  to  it,  and  consequently  are 
equal  to  each  other.  And  so  of  all  the  particular  exchange- 
values  of  any  one  thing. 

§  4.  To  measure  even  the  particular  exchange-value  of  one 
thing  in  another  is  not  an  easy  task  ;  for  at  the  same  place 
during  the  same  period  of  time  the  same  quantity  of  the  same 
kind  of  thing  may  be  exchanged  for  various  quantities  of  an- 
other kind  of  thing.  It  is  sometimes  said  that  the  exchange- 
value  of  one  thing  in  another  is  determined  by  an  actual  ex- 
change. This  is  not  so,  as  it  often  happens  that  in  an  actual 
exchange  the  one  party  rejoices  over  a  good  bargain  and  the 
other  is  worried  lest  he  have  made  a  bad  bargain — L  e.,  the  one 
thinks  he  has  got  more,  the  other  less,  than  the  thing  given 
was  worth.  We  also  ascribe  exchange-value  to  things  never 
exchanged,  and  especially  we  w^ant  to  know  the  proper  ex- 
change-value of  a  thing  which  we  are  going  to  part  with,  before 
we  effect  its  exchange  for  anything  else.  We  estimate  ex- 
change-values rather  by  the  general  run  of  exchanges  of  simi- 
lar other  things.  The  particular  exchange-value  of  one  kind 
of  thing  in  another  kind  of  thing  is  not  an  affair  of  a  single 
exchange,  but  of  many.  The  single  exchanges  may  fluctuate 
around  an  average,  which  is  what  we  call  the  exchange-value 
of  the  thing  (the  class)  during  the  period  in  question.  When 
certain  kinds  of  things  are  habitually  exchanged  in  large  quan- 
tities, the  fluctuations  in  short  intervals  of  time  are  not  apt  to 


20  THE  NATURE  OF  EXCHANGE- VALUE 

be  large,  so  that  in  these  cases  it  is  tolerably  easy  to  determine 
their  particular  exchange-values  within  narrow  limits  of  error. 

§  5.  If  the  particular  exchange-value  of  one  thing  in  another 
is  sometimes  difficult  to  estimate,  it  is  always  difficult  to  esti- 
mate its  general  exchange-value  ;  for  this  includes  not  only  the 
determination  of  its  many  particular  exchange-values,  some  of 
which  are  sure  to  be  troublesome,  but  also  the  combination  of 
these  into  a  whole.  This  problem,  however,  may  perhaps  ad- 
mit of  a  satisfactory  theoretical  solution,  whereupon  the  diffi- 
culties will  reside  only  in  the  practical  details. 

The  labor  of  finding  the  particular  exchange-values  of  things 
would  be  almost  infinite  if  we  were  to  undertake  to  find  all  the 
particular  exchange-values  of  all  things.  For  as  one  thing  has 
a  particular  exchange-value  in  another  thing,  this  has  a  par- 
ticular exchange-value  in  it,  so  that  there  are  two  relations  of 
particular  exchange-value  between  every  two  things ;  and, 
therefore,  among  a  hundred  kinds  of  things,  each  one  of  the 
hundred  having  a  particular  exchange-value  in  every  one  of  the 
other  ninety  and  nine,  there  would  be  100  x  99  =  9,900  par- 
ticular exchange- value  relations,  and  among  two  hundred  there 
would  be  200  x  1  99  =  39,800,  and  so  on  in  rapidly  increasing 
progression.  Through  this  maze  of  interminable  interrelations 
we  have  been  enabled  to  make  our  way,  as  is  well  known,  by  the 
invention  of  money,  which,  as  is  said,  serves  as  the  "  common 
denominator  "  for  all  other  things.  By  means  of  prices,  money 
acts  as  a  perfectly  satisfactory  measure  of  the  exchange-values 
of  other  things  in  the  same  place  at  the  same  time.  Now  money 
alone  is  brought  into  direct  relationship  with  every  other  ex- 
changeable thing,  the  relations  between  these  others  being  inter- 
mediated through  their  relations  to  money.  Hereby  our  atten- 
tion is  confined  to  the  particular  exchange-values  of  all  other 
things  in  money,  as  indicated  by  their  prices,  or,  which  is  the 
same  thing  inverted,  to  the  particular  exchange-values  of  money 
in  all  other  things  singly.  Consequently  it  is  primarily  only 
the  general  exchange-value  of  money  in  all  other  things  col- 
lectively that  we  arc  concerned  with  measuring ;  for  after 
measuring  it  and  finding  its  constancy  or  variation  at  different 


THE    QUAXTITATIVEXESS    OF    EXCHAXGE-VALUE  21 

times,  or  in  different  places,  we  can  measure  the  constancy  or 
variation  of  any  other  thing  in  general  exchange-value  by  its 
known  constancy  or  variation  in  relation  to  money.  It  is  not 
an  uncommon  opinion  that  it  is  easier  to  estimate  the  variation 
in  exchange-value,  simply  spoken,  of  any  commodity  than  it  is 
to  estimate  the  variation  in  the  exchange-value  of  money,  on 
the  ground  that  in  the  former  case  we  only  have  to  notice  the 
variation  of  its  price,  while  in  the  latter  we  have  to  consider 
variations  of  all  prices.  But  the  variation  of  a  commodity's 
price  only  tells  the  variation  of  the  commodity's  particular 
exchange- value  in  money,  and  gives  us  no  information  concern- 
ing the  variation  or  constancy  of  the  commodity's  exchange- 
value  simply  so  called,  or  general  exchange-value,  until  we 
know  the  variation  or  constancy  of  the  general  exchange-value 
of  money.  Thus  the  former  calculation  really  presupposes  the 
latter.  And  as  a  matter  of  fact,  not  only  is  it  easier  to  meas- 
ure the  constancy  or  variation  of  the  general  exchange-value  of 
money  than  of  anything  else,  but  money  is  the  only  thing  of 
which  the  general  exchange-value  can  be  measured  by  us  di- 
rectly ;  for  we  should  never  be  able  to  find  all  the  particular 
exchange-values  of  any  other  thing  without  taking  account  of 
its  and  of  the  other  things'  prices,  or  exchange-values  in 
money. 

§  6.  Because  exchange- value  is  quantitative,  to  conceive  of 
exchange-value  involves  measurement.  The  measurement  may 
be  rough  or  exact,  but  measurement  there  must  be.  And  we 
make  measurement  not  only  of  particular  exchange-values,  but 
also  of  general  exchange-value — especially  of  money's  general 
exchange-value.  Everybody  has  some  notion  of  "  money's 
worth,"  and  some  opinion  as  to  whether  through  a  course  of 
years  this  worth  or  exchange-value  has  remained  constant  or 
varied,  or  whether  it  is  greater  in  one  place  than  in  another. 
Only  this  notion  is  generally  very  badly  formed  and  left  vague 
and  unprecise,  wherefore  the  opinion  is  generally  weak  and  ir- 
resolute, or  inclines  in  favor  of  constancy  merely  through  lack 
of  proof  of  variation.  Political  economy,  if  it  be  a  science, 
cannot  avoid  the  duty  of  attempting  to  rid  this  notion  of  its 


,22  THE  NATURE  OF  EXCHAXdE- VALUE 

vagueness  ^  and  to  provide  a  method  of  measuring  the  general 
exchange-value  of  money  with  theoretical  precision  as  a  model 
to  be  realized  as  closely  as  possible  in  practice.* 

IV. 

§  1.  In  measuring  (juantities  we  must  bear  in  mind  that  we 
are  not  concerned  with  the  causes  of  their  constancy  or  of  their 
variations.  In  measuring  from  year  to  year  the  weight  and 
tallness  of  a  boy,  we  have  nothing  to  do  with  the  causes  that 
make  him  grow\  To  measure  variations,  and  to  explain  them 
by  pointing  out  their  causes,  are  two  distinct  operations.^  And 
the  former  is  the  primary  ;  for  we  can  be  scientifically  prepared 
for  investigating  the  cause  of  variations  only  after  measuring  the 
variations  with  scientific  precision.  When  direct  measurement 
of  things  we  desire  to  measure  is  not  feasible,  Ave  frequently 
measure  them,  as  less  apparent  causes,  by  their  more  apparent 
effects,  if  we  can  eliminate  all  other  causes,  as  in  the  familiar 
example  of  heat,  by  expansion.  But  less  apparent  effects  w^e 
rarely  attempt  to  measure  by  their  apparent  causes,  since  these 
effects  generally  escape  our  control  and  we  cannot  know  whether 
they  are  operated  upon  only  by  the  causes  employed  to  measure 
them  by.  And  equally  apparent  effects  it  would  be  useless  to 
measure  by  their  equally  apparent  causes,  as  we  can  measure 
them  directly.  To  attempt,  then,  to  measure  apparent  effects 
by  their  less  apparent  causes,  would  be  the  height  of  folly. 
Now  the  variations  of  exchange- values  are  more  apparent  than 

'J.  S.  Mill  spoke  of  "the  necessary  indefiniteness  of  the  idea  of  general  ex- 
change-value," op.  ciL,  Vol.  II.,  p.  102.  Of  course  until  it  is  made  definite, 
this  idea  will  remain  indefinite.  But  what  shows  the  necessity  of  its  remaining 
indefinite?  And  why  should  the  task  of  clarifying  it  l)e  shirked  by  any  econo- 
mist? 

*  J.  I>.  Say  said  we  cannot  "  measure"  exchange-value  at  difi'crent  times  and 
places,  but  wecan  only  "  appraise  "  it,  or  form  "  approximative  valuations,"  be- 
cause of  the  absence  of  an  invariable  measure — /.  e.,  because  we  cannot  make  an 
absolutely  exact  measurement,  op.  cit.,  Vol.  II.,  pp.  85,  93.  This  distinction  is 
too  hard  and  fast,  since  an  inexact  measurement  is  a  measurement,  and  if  we 
took  his  statement  literally,  we  should  have  no  measures,  as  none  is  absolutely 
invariable.  We  ought  at  least  to  aim  at  more  than  appraising  ;  we  ought  to  aim 
at  measuring. 

1  Cf.  Jevons,  B.  22,  pp.  21,  59 ;  Nicholson,  B.  94,  pp.  :i04-305. 


MENSUIIATIOX    NOT    CONCERNED    WITH    CAUSES  23 

their  causes ;  for,  difficult  tiiough  it  })e  to  measure  accurately  a 
variation  in  the  general  exchange-value  of  money,  it  is  less 
difficult  than  to  assign  all  the  exact  causes  which  have  produced 
it  (although  it  may  be  easier  to  adduce  many  possible  causes  in 
a  general  way  than  actually  to  measure  the  precise  variation). 
Therefore  it  is  especially  absurd  in  political  economy  to  say 
that  in  attempting  to  measure  exchange-value  we  must  pay  at- 
tention to  its  causes.  On  the  contrary,  we  should  be  especially 
careful  to  drop  all  consideration  of  its  causes. 

§  2.  It  is  necessary  to  state  this  obvious  truth  because  of  the 
prevalence  of  the  opposite  opinion  in  this  one  subject  alone 
among  all  subjects  of  metrology.  Thus,  for  example,  a  writer 
has  recently  asserted  :  "  To  measure  the  variations  of  money 
there  is  required  not  only  detailed  knowledge  of  prices,  but 
also  of  the  causes  which  produce  the  variations." '  And 
another  has  said  that  this  measurement  is  impossible  because 
of  the  impossibility  of  knowing  all  these  causes.^  It  would  be 
difficult  to  match  such  assertions  with  similar  assertions  in  any 
outer  branch  of  science.'*  Yet  opinions  of  this  sort  in  political 
economy  have  no  less  an  authority  than  that  of  Ricardo.^ 

The  reason  for  this  error  is  twofold.  It  lies  in  the  confusion 
between  cost-value  and  exchange- value,  and  in  the  doctrine  that 
"  value,"  including  exchange-value,  is  determined  by  the  labor- 
cost  of  production.  But  it  is  only  cost-value  which  is  propor- 
tionate to  labor-cost;  and  even  if  exchange-values  were  de- 
termined by  the  relative  labor-costs,  it  would  not  follow  that 
the  exchange-value  of  a  thing  can  be  measured  by  its  labor- 
cost,  but  only  by  its  labor-cost  compared  with  the  labor-costs 
of  other  things,"  which   comparison   would  be    more   difficult 

2  Nitti,  La  misura  delle  variazioni  di  valore  delta  moneta,  1895.  (Quoted  from 
the  Economic  Journal,  Vol.  V.,  p.  260.)  Of.  V.  Pareto,  Cours  d' economic  poli- 
tique, Lausanne  180(3,  Vol.  I.,  p.  200. 

^  Martello,  op.  cit.,  p.  3;33. 

*  On  the  contrary,  e.  g.,  Whewell  speaks  of  "  an  important  maxim  of  induc- 
tive science,  that  we  must  first  obtain  the  measure  and  ascertain  the  lairs  of  phe- 
nomena, before  we  endeavor  to  discover  their  causes,"  Philosophy  of  the  inductive 
sciences,  1847,  Vol.  II.,  p.  240. 

5  Works,  pp.  400-401. 

^  As  long  ago  clearly  pointed  out  by  R.  Torrens,  Essay  on  the  produciton 
of  wealth,  London  1821,  pp.  49,  50,  and  by  S.  Bailey,  Critical  dissertation  on 
the  nature,  measure,  and  causes  of  value,  London  1825,  pp.  6-11,  17-18,  and 
Letter  to  a  political  economist,  1820,  pp.  53-54. 


24  THE    NATURE    OF    EXCHANGE-VALUE 

than  the  comparison  between  their  actual  ratios  of  exchange, 
and  so  would  be  worthless  even  if  that  doctrine  were  univer- 
sally true,  and  is  especially  M'orthless  since  that  doctrine  is  not 
universally  true  (and,  in  fact,  never  even  pretended  to  be). 

§  3.  Another  objection  is  a  variation  upon  the  same  theme. 
This  is  that  the  measurement  of  the  general  exchange-value 
of  money,  however  successfully  made,  is  useless,  because  it 
gives  no  information  about  the  causes  of  the  variations."  It 
may  seem  strange  that  any  economists  could  raise  such  an  objec- 
tion ;  for  in  no  subject  does  a  measurement  disclose  causes,  and 
in  nothing  else  is  mensuration  reproached  for  its  inability  to  do 
so.  The  explanation  is  that  what  these  economists  want  is 
really  a  measurement  of  cost-value,  and  so  they  are  dissatisfied 
with  what  turns  out  to  be  a  measurement  of  exchange-value 
alone,  which  being  called  simply  a  measurement  of  "  value," 
may  have  seemed  to  them  to  give  promise  of  being  a  measure- 
ment also  of  cost-value.*^  What  they  want  is  a  measurement 
of  the  variations  in  the  cost-values  of  commodities  and  in 
the  cost-value  of  money  (gold) — or  rather,  a  measurement  of 
the  variations  in  money  as  a  standard  of  cost- value.  Or  even 
sometimes  they  want  such  a  measurement  of  money  as  a  stand- 
ard of  esteem-value ;  for  some  of  them  recognize  that  gold  is 
a  semi-monopolized  product,  with  "  value  "  enhanced  by  rarity 

■^  So  D.  A.  Wells,  Recent  economic  changes,  1889,  p.  121.  Cf.  Malthus,  op.  cit., 
p.  120,  and  McCulloch,  Political  economy,  p.  214,  (Note  to  Wealth  of  nations, 
pp.  439-440). 

*  Malthus  offers  a  peculiar  example  of  confusion,  lie  wrote  :  "  The  exchange- 
able value  of  a  eoininodity  can  only  he  proportioned  to  its  general  power  of 
purchasing  [general  exchange-value]  so  long  as  the  commodities  with  which  it 
Is  exchanged  continue  to  be  obtained  with  the  same  facility,"  op.  cit.,  pp.  58-59. 
Tims  although  he  expressly  uses  the  term  "exchangeable  value,"  he  distinguishes 
it  from  purchasing  power  or  exchange-value  proper,  and  identifies  it  with  cost- 
value  He  does  so  still  more  plainly  when  he  amplifies  the  term  into  "intrinsic 
value  in  exchange,"  p.  CO.  Yet  the  idea  of  exchange-value  always  attaches  to 
these  terms,  being  embodied  in  them.  The  fundamental  fault,  which  runs  through 
all  hislongdisiiuisition  on  the  measure  of  value,  and  prevents  it  from  reaching  a  sat- 
isfactory conclusion,  is  file  fact  that  he  is  seeking  the  impossible — a  single  measure 
both  of  exchange;- value  and  of  cost-value. — llicardo  had  the  same  ideaasMaltlius 
when  he  wrote:  "  AVhy  should  .  .  .  all  commodities  together  be  the  standard, 
when  such  a  standard  is  itself  subject  to  fluctuations  in  value  ?  Works,  p.Kiii,  i.  e., 
in  cost-value  (and  in  esteem-value).  But  of  course  their  fluctuation  in  other 
kinds  of  value  is  not  a  reason  why  they  should  not  be  the  standard  of  exchange- 
value  if  they  be  found  not  to  fluctuate  in  this. 


MENSURATION    NOT    CONCERNED    WVrU    CAUSES  25 

above  its  labor-cost — that  is,  with  esteem-value  greater  than  its 
cost-value.  And  this  measurement  they  think  can  be  made  by 
examining  the  variations  in  the  costs  of  production  of  commo- 
dities and  allowing  for  them  in  the  variations  of  prices  ;  for 
they  consider  that  the  causes  of  the  variations  of  prices  lie  to 
a  great  extent  in  the  altered  costs  of  production  of  the  com- 
modities themselves,  leaving  at  times  a  remainder,  which  is  the 
sole  cause  of  the  variation  of  prices  due  to  the  influence  of 
money,  and  consequently  the  only  measure  of  the  "  inner  " 
variation  of  the  "  value  "  of  money,  or  of  its  appreciation  or 
depreciation  f — all  which  may  be  correct  enough,  but  only  of 
the  cost-value  of  money,  or  of  its  esteem-value,  and  not  of  its 
exchange-value.'" 

Now  these  economists  have  a  perfect  right  to  their  wish,  and 
if  they  succeeded  in  inventing  a  method  by  which  the  constancy 
or  variation  of  the  monetary  standard  in  cost- value  or  in  esteem- 
value  could  be  measured,  they  would  be  rendering  an  important 
service  in  political  economy.  But  there  is  room  in  political 
economy  for  the  measurement  of  exchange-value  as  well  as  for 
the  measurement  of  cost-value  or  of  esteem-value.  If  political 
economy  be  at  all  a  science  of  exchanges,  its  need  for  a  meas- 
urement of  exchange-value  is  very  great,  and  undiminished  by 
any  further  want  which  may  be  felt  for  a  measurement  of  other 
kinds  of  value.  Let  us  then  pursue  our  course  of  seeking  the 
method  of  measuring  general  exchange-value,  undeterred  by 
the  fact  that  this  measurement  is  not  a  measurement  of  some- 
thing else. 

^  Cf.  W.  Lexis  in  the  Verhandlungen  der  deutschen  Silbcrkomm-ission,  1894, 
p.  153;  K.  Ilelfierich  Die  Wahrungsfrage,  1895,  p.  18.  Probably  this  was  the 
opinion  also  of  llicardo  in  the  passage  referred  to  in  Note  5. 

1"  They  even  then  object  to  the  measurement  of  exchange-value  as  being  mis- 
leading (so  especially  G.  M.  Flamingo,  The  measure  of  the  value  of  money  accord- 
ing to  European  economists,  Journal  of  Economics,  Chicago,  Dec.  1898,  pp.  74- 
75)  for  a  reason  which  amounts  to  this  :  that  people  may  mistake  a  conclusion 
concerning  exchange-value  for  a  conclusion  concerning  cost- value  or  esteem- value. 
But  the  best  way  for  them  to  keep  others  from  such  error  is  for  themselves  to 
avoid  the  confusion  of  thought  between  the  several  kinds  of  value. 


CHAPTER   II. 

THE   CORRELATION   OF   EXCHANGE-VALUES. 

I. 

§  1.  The  luutual  exchange-values  of  two  things,  each  in  the 
other,  are  subject  to  a  very  simple  law.  If  A  rises  in  ex- 
change-value in  B,  ijjso  facto  B  falls  in  exchange-value  in  A ; 
and  reversely.  A  variation  of  one  tiling^ s  exchange-value  in  an- 
other is  an  opposite  variation  of  the  latter's  exchange-value  in  the 
former  (Proposition  VIIL).  Of  two  boys  at  one  time  evenly 
tall,  if  the  one  grows  taller  than  the  other,  the  other  ipso  facto 
becomes  shorter  than  the  first.  Here  we  are  apt  to  say  the 
second  boy  has  not  grown  shorter,  but  may  even  have  grown 
taller,  only  the  first  has  grown  more.  We  do  so  because  we 
have  other  things  in  mind,  with  the  lengths  of  which  we  com- 
pare these  lengths  as  well  as  with  each  other.  If  the  boys 
were  in  space  alone  and  it  were  impossible  to  compare  their 
lengths  with  any  other  lengths,  a  change  in  their  relative 
lengths  would  enable  them  only  to  say,  "  You  have  become 
taller  than  I "  and  "  You  have  become  shorter  than  I." 

It  may  be  said  that  if  we  knew  in  which  of  the  boys  the 
cause  of  the  change  lay,  we  should  then  know  which  of  the 
boys  has  changed  and  which  not  (or  how  much  each) — in 
something  called  absolute  length.  But  it  is  evident  that  we 
cannot  seek  for  the  cause  of  a  change  until  we  know  the  change, 
and  so  we  must  know  which  of  the  boys  has  changed  before  we 
can  know  where  the  cause  lies.  Similarly  with  the  case  of  ex- 
change-value. Compared  with  the  exchange-values  of  other 
things,  and  with  a  view  to  the  variations  of  their  exchange- 
values  in  other  things,  the  change  between  A  and  B  may  be  a 
change  only  in  the  one,  or  even  both  may  have  changed  in  the 

26 


BETWEEN    TAVO    THINGS  27 

same  direction,  but  the  oue  more.  Here  it  has  actually  been 
said,  Avith  much  reiteration,  that  without  regard  to  other  things, 
it  is  sufficient  for  us  only  to  know,  or  it  is  necessary  for  us  to 
know,  on  which  side  the  cause  of  the  change  lay,  and  then  we 
should  know  which  of  the  two  has  changed — in  "  value,"  still 
appearing  to  mean  exchange-value,  now  of  some  absolute  sort 
(except  that  we  know  that  these  writers  associate  even  the 
term  "  value  in  exchange  "  with  value  of  other  kinds). ^  The 
cause  in  this  matter  has  generally  been  sought  in  the  labor  of 
producing  the  article  (though  it  might  equally  well  be  sought 
in  the  quantity  of  the  article  forthcoming,  relatively  to  the 
number  of  people  desiring  it) ;  and  so  it  is  said  that  if  we  knew 
that  this  labor  (or  this  quantity)  has  changed  in  the  case  of  one 
of  the  articles  only,  we  should  know  which  of  them  alone  has 
changed  in  "  value " — supposedly  meaning  exchange-value. 
But  if  so,  the  statement  is  wrong.  For,  under  the  supposition, 
we  know  nothing  about  other  things  ;  and  so  if  we  knew  only 
of  such  a  change  in  the  labor-cost  (or  in  the  quantity)  of  one  of 
the  articles,  we  should  only  know  that  its  cost-value  (or  esteem- 
value)  has  changed,  not  that  its  exchange-value  has  changed — 
by  "  exchange-value "  properly  meaning  only  exchange-value 
simply  so  called,  that  is,  exchange-value  in  all  other  things  ; 
for  the  rest  are  unknown,  or  left  out  of  account.  But  as  re- 
gards its  exchange-value  in  the  one  other  thing,  we  know  that 
this  has  changed  exactly  as  we  know  that  the  other's  exchange- 
value  in  it  has  changed,  learning  both  from  actual  exchanges ; 
and  we  know  these  changes  just  as  well  whether  we  know 
the  changes  in  their  costs  of  production  (or  quantities),  or  not. 
In  considering  the  mutual  exchange-values  of  two'  things  alone, 
we  are  abstracting  all  other  things  ;  and  now  the  only  change 
we  have  a  right  to  consider  is  the  change  between  them,  which 
involves  both  a  rise  and  a  fall. 

§  2.  In  thus  putting  aside  all  other  things  and  abandoning 
the  use  of  them  as  a  common  measure,  we  can  measure  the  ex- 
change-value in  other  things  of  each  of  the  two  objects  only  by 

^  U.  g.,  J.  E.  Cairnes,  Some  leading  principles  of  political  economy,  1874,  pp. 
13-14.  So  even  Walras,  B.  71,  pp.  5-0. — Shadwell  disposes  of  this  question  by 
pronouncing  it  "puerile,"  op.  cit.,  p.  103. 


28  THE    CORRELATION    OF    EXCHANGE- VALUES 

the  other.  As  we  employ  a  different  measure  for  each,  the  rise 
and  the  fall  are  not  found  to  be  in  the  same  proportion.  Evi- 
dently if  A,  after  being  equivalent  to  B,  rises  by  one-half,  or 
by  50  per  cent.,  in  exchange-value  in  B,  so  that  A  exchanges 
for  l^B,  then,  1|B  exchanging  for  lA,  IB  exchanges  for  |A, 
wherefore  B  has  fallen  by  one-third,  or  by  33^  per  cent.,  in 
exchange-value  in  A. 

The  relationship  just  exemplified  may  be  formulated  and 
generalized  as  follows.  Here  and  throughout  these  pages 
whenever  percentage  is  represented  in  algebraic  formulae  by  the 
sign  p  or  the  like,  this  refers  to  the  percentage  on  1,  or  to 
percentage  expressed  in  hundredths,"  Xow,  then,  if  A,  after 
being  equivalent  to  B,  rises  p  per  cent,  in  exchange-value  in  B, 
so  that  it  commands  (1  +j9)B,  and  has  risen  to  (1  -f- p)  times 
its  former  exchange-value  in  B,  then  B,  now  exchanging  for 

A,  has  fallen  by  1  —  -^ — , —  or  z. per  cent,  to 


-.  —  1  or 7  per  cent,  to  q ;  times  its  former  ex- 

—  //  1  —  //  ^  1  —  » 


1  +  i>    '  '  1  +  i?        1  +  ^3  ^  ■         1  +  p 

of  its  former  exchange-value  in  A.     Reversely  if  A,  after  being 

equivalent  to  B,  falls  p'   per  cent,  to  (1  — p')^,  B  rises  by 

1  p'  1 

-    ;  —  1  or 7  per  cent,  to  :, , 

1  —  p  1  —p    ^  1  —p 

change-value  in  A.  Thus  the  exchange-value  of  A  in  B  and 
of  B  in  A  being  in  each  instance  rep resen table  at  first  by  1,  we 
find  that  unity  is  the  geometric  mean  between  the  later  exchange- 
value  of  A  in  B  and  the  later  exchange-value  of  B  in  A ;  for 

1 
l+p' 


1-f-p:  1  ::  1 
and 


\-p'  :  1  ::  1 


1  -y 

This  relationship  may  be  stated  thus  :  When  qffico  equivalents  the 
one  rises  or  falls  in  exehanye-vaJue  in  the  oilier,  the  other  fa  lis  or  rises 

-  If  the  popular  form  be  desired,  2^  referring  to  integral  figures,  all  the  formulae 
which  follow  may  be  maintained  by  substituting  100  wherever  1  occurs  or  is 
understood.  IJ.  g.,  here  100  A,  after  being  equivalent  to  100  B,  rising  ^j  per  cent., 
command  (100+^)  B,  etc.    And  always  the  expression  "  {\  +  p)  times"  must 

be  changed  to  "~Tr,7j —  times."     It  will  be  seen  that  the  method  adopted  is  the 
8inii)U'r. 


BETWEEN    TWO    THINGS  29 

in  exchange-value  in  it  so  that  their  subsequent  mutual  exchange- 
values  are  reciprocals  of  each  other,  the  quantities  in  which  they 
first  exchange  being  taken  as  units.  Or,  again,  more  briefly,  the 
rise  of  the  one  measured  in  the  other  is  the  inverse  of  the  fall  of  the 
latter  measured  in  the  former ;  and  reversely  (Proposition  IX.). 

We  may,  of  course,  in  our  supposition  replace  B  by  M, 
representing  a  sum  of  money,  say  one  unit.  The  same  rule 
holds,  but  may  be  differently  worded.  Price  is  the  expression 
of  the  exchange-value  in  money  of  the  article  priced.^  If  A 
rises  in  price  from  1.00  to  1.50,  its  exchange- value  in  M  rises 
in  direct  proportion,  and  the  exchange-value  of  M  in  it  falls  in 
inverse  proportion,  M  now  purchasing  only  so  much  of  A  as 
.66fM  previously  purchased.  The  rise  of  A  in  price  by  50 
per  cent.,  is  a  depreciation  of  money  in  A  by  33^  per  cent. 
Reversely  if  the  price  of  A  falls  from  1.00  to  .50,  or  by  50  per 
cent.,  the  exchange-value  of  M  in  A  has  risen  by  100  per 
cent.,  M  now  purchasing  2A.  The  price  being  originallij  one 
unit,  the  subsequent  price  and  the  subsequent  exchange-value  of 
money  in  the  article  priced  {expressed  in  its  original  exchange- 
value  in  it)  are  reciprocals  of  each  other  (Proposition  X.). 

§  3.  In  these  reciprocal  changes  it  is  possible  to  affirm  that,  in 
spite  of  the  difference  in  the  proportions,  which  arises  from  the 
differences  in  the  measures  used,  the  rise  of  the  one  thing  in  the 
other  and  tJie  corresponding  fall  of  the  latter  in  the  former  are 
equal  (Proposition  XL).  A  rise  of  A  to  double  its  former  ex- 
change-value in  B,  for  instance,  and  the  fall  of  B  to  half  its 
former  exchange-value  in  A,  are  equal  changes.  For  the  ex- 
change-value of  A  in  B  rises  from  the  power  possessed  by  1 A  of 
purchasing  IB  to  that  of  purchasing  2B  ;  and  if  it  should  fall 
again  from  this  power  to  that  of  purchasing  IB,  this  fall  would 

3  Economists  have  not  been  careful  in  their  use  of  the  term  "price,"  defining 
it  indifferently  in  two  distinct  ways:  (1)  as  the  sum  of  money  given  (or  aslied) 
in  exchange  for  the  thing,  (2)  as  the  value  of  the  thing  in  money — and  sometimes 
the  same  writer  has  given  both,  e.  g.,  J.  S.  Mill,  op.  cit..  Vol.  I.,  p.  538  ;  J.  Garnier, 
Traite  d'eco)wmie  politique,  1848,  9th  ed.,  1889,  p.  293 ;  Macleod,  Elements  of 
political  economy,  1858,  p.  39  and  Theory  of  credit.  Vol.  I.,  p.  176.  The  first  is 
more  conformable  to  popular  usage,  and  is  the  best.  Price  is  not  the  exchange- 
value  of  the  thing  in  money,  but,  being  the  sum  of  money  itself,  it  expresses  or 
measures  the  exchange-value  of  the  thing  in  money. 


30  THE    CORRELATION    OF    EXCHANGE- VALUES 

evidently  be  equal  to  that  rise.  In  the  original  variation  the 
exchange-value  of  B  in  A  falls  from  the  power  possessed  by 
2B  of  purchasing  2A  to  that  of  purchasing  lA.  Tliis  fall 
is  evidently  equal  to  the  last  mentioned  fall  of  A,  and  conse- 
quently to  that  rise  of  A.  In  effect,  a  rise  from  1  to  2  and  a 
fall  from  2  to  1  are  equal  to  each  other. 

§  4.  Whenever  such  mutual  changes  take  i^acc  between  two  ar- 
ticles relatively  to  each  other,  their  exchange-values  in  all  things 
beside  themselves  vary  in  the  same  jyroportions  relatively  to  each 
other  (Proposition  XII.).  When  A  and  B  are  equivalent  to 
each  other,  we  have  seen  that  they  possess  the  same  exchange- 
value  in  all  other  things,  whatever  this  exchange-value  may  be. 
Then  if  A  rises  50  per  cent,  in  its  particular  exchange-value 
in  B,  it  is  evident  that,  iu  relation  to  all  the  other  things  which 
are  outside  them  both,  A's  exchange-value  in  all  those  other 
things  has  risen  50  per  cent,  above  B's  exchange-value  in  all 
those  other  things,  whatever  this  exchange-value  of  B  may  be  ; 
and  inversely  B's  exchange-value  in  all  those  other  things  has 
fallen  33^  per  cent,  below  A's  exchange-value  in  all  those  other 
things,  whatever  this  exchange-value  of  A  may  be. 

What  has  just  been  said  is  not  to  be  said  of  the  exchange- 
value  of  A  in  all  other  things,  which  would  include  B,  because 
B  has  changed  relatively  to  A,  and  so  may  be  a  disturbing 
factor ;  nor  of  the  exchange-value  of  B  in  all  other  things, 
which  would  include  A,  because  A  has  changed  relatively  to 
B,  and  so  may  be  a  disturbing  factor ;  wherefore  the  relation- 
ship between  these  exchange-values  is  slightly  different.  But 
the  disturbance  caused  by  B's  presence  in  A's  general  exchange- 
value  is  counteracted  if  A  also  is  included ;  similarly  in  the 
case  of  B's  general  exchange-value,  if  B's  presence  is  also  in- 
cluded, wherefore  the  above  relationship  is  restored.  That  is, 
when  mutual  changes  take  lolace  between  two  tilings,  relatively  to  each 
other,  their  exchange-values  in  all  things  vary  in  the  same  'propor- 
tions relatively  to  each  other  (Proposition    XIII.).''      Another 

■*  In  practice,  of  course,  the  variation  in  their  exchange-values  in  all  other 
things  will  be  almost  the  same  as  their  variations  in  all  things, — as  will  be  shown 
later.  The  Proposition,  then,  is  approximately  true  of  the  "general  purchasing 
power"  of  two  things. 


BETWEEN    TWO    THINGS  31 

demonstration  of  this  proposition  will  be  forthcoming  when 
we  have  discovered  how  to  measure  exchange-value  in  all  things. 

It  follows,  of  course,  that  if  two  things  (classes)  remain  un- 
changed 7'elatively  to  each  other,  their  exchange-values,  both  in  all 
the  common  other  things  and  in  all  things,  vary,  or  remain  con- 
stant, alike  (Proposition  XIV.) ;  but  not  necessarily  do  their 
exchange-values  in  all  other  things  vary  alike. 

Naturally  ivhen  mutual  changes  take  place  between  two  things, 
some  change  must  take  place  in  one  or  in  both  relcdively  to  the 
common  other  things  (or  relatively  to  all  things,  and  also  to  all 
other  things)  ^  (Proposition  XV.).  The  exchange- values  of  the 
two  things  in  all  the  common  things  (or  in  all  things)  are 
yoked  together  so  that,  the  variation  of  the  one  in  the  other 
being  given,  and  also  the  variation  (or  constancy)  of  either  in 
all  the  others  (or  in  all  things),  the  variation  (or  constancy) 
of  the  other  in  all  the  others  (or  in  all  things)  may  be  easily 
calculated.  In  general  the  possible  changes  may  be  classitied 
into  jive principcd  types:  relatively  to  the  other  things  (1)  A 
may  have  risen,  while  B  has  remained  unchanged,  (2)  A  may 
have  remained  unchanged,  while  B  has  fallen,  (3)  A  may  have 
risen  and  B  fallen,  (4)  both  A  and  B  may  have  risen,  but  A 
more,  (5)  both  A  and  B  may  have  fallen,  but  B  more.  Mere 
knowledge  that  A  has  risen  in  B,  with  or  without  the  percentage 
being  known,  con\'oys  no  knowledge  as  to  which  of  these  pos- 
sible combinations  of  changes  relatively  to  the  other  things  has 
been  effected  ;  but  it  does  convey  knowledge  that  one  or  another 
of  them  must  have  been.  Which  of  them  it  is,  or  exactly  what 
the  changes  are  relatively  to  all  the  common  other  things  (or  to 

*  This  Proposition,  therefore,  is  true  of  the  general  purchasing  power  of  two 
things.  Yet  the  following  statement  has  been  made: — "Whilst  gold  has  not 
risen  in  purchasing  power,  and  silver  has  not  declined  in  purchasing  power,  the 
relative  value  of  silver  to  gold  has  declined  to  less  than  half,"  A.  Ellissen,  The 
errors  and  fallacies  of  bimetallism,  London  1895,  p.  12.  The  author,  however,  is 
conscious  of  inconsistency,  for  he  adds :  "  In  mathematics  such  a  problem  would 
amount  to  a  preposterous  absurditj^ ;  but  in  the  case  before  us  it  is  only  too  true." 
It  is  tenable  only  by  an  equivocation,  the  author  having  treated  "purchasing 
power"  like  "value"  in  general,  and  then  not  having  distinguished  between  its 
different  species.  He  means  that  silver  has  not  fallen  in  purchasing  power  proper 
and  gold  has  not  risen  in  cost-value  (or  esteem-value) — perfectly  simple  and  un- 
connected statements,  not  deserving  to  be  so  expressed  as  to  involve  contradiction. 


32  THE    CORRELATION    OF    EXCHANGE-VALUES 

all  things)  must  be  sought — not  by  trying  to  find  the  causes,  or 
changes  in  the  costs  of  production  (or  in  the  quantities)  of  either 
or  both,  or  of  all  the  other  things,''  but — by  direct  measure- 
ment either  of  A's  or  of  B's  exchange- value  in  all  the  other 
things  (or  in  all  things),  or  of  both,  which  measurement  must 
take  account  of  all  the  particular  exchange-values  of  either  or 
each  in  all  the  other  things  singly,  and  thereby  also  of  the  par- 
ticular exchange- values  of  all  the  other  things  in  either  or  each 
of  these  two. 

II. 

§  1.  Let  us  now  pay  attention  to  the  first  and  the  second, 
as  the  simplest,  of  the  above  five  possibilities  when  the  other 
things  are  also  taken  into  consideration.  Let  us  suppose  that, 
A  rising  or  falling  in  exchange-value  in  B,  B  retains  its  ex- 
change-value unchanged  relatively  to  all  the  other  things,  and 
that  these  other  things  all  remain  without  change  relatively  to 
one  another  and  to  B.  Now,  then,  if  A  rises  in  exchange- 
value  in  B,  it  rises  also  in  exchange-value  in  every  one  of  the 
other  things,  that  is,  in  every  other  thing  beside  itself,  in  the 
same  proportion  ;  and,  B  ipso  facto  falling  in  exchange-value 
in  A,  so  does  every  one  of  the  other  things  fall  in  exchange- 
value  in  A,  and  in  the  same  proportion  as  B,  that  is,  in  inverse 
proportion  to  A's  rise  in  B.  Thus  if  A  rises  50  per  cent,  in 
B,  it  rises  50  per  cent,  in  C,  in  D,  and  so  on  throughout  all  the 
otiier  things  ;  and  as  B  falls  33J  per  cent,  in  A,  so  do  C  and 
D  and  the  others,  each  singly,  fall  3o^  per  cent,  in  A.  And 
the  reverse  is  the  case  if  A  fiills  in  B.  Then,  like  B,  all  the 
otlicrs  rise  singly  in  A  in  inverse  proportion  to  A'«  fall  in 
B.  Thus  we  have  the  general  law  :  A  variation  of  one  thing  in 
exchange-value  in  the  same  proportion  relatively  to  every  other  thing 
involves  an  opposite  variation  in  inverse  proportion  of  every  one 
of  the  others  relatively  to  it  (Proposition  XVI. ). 

^  These  are  subjects  tliat  must  be  investigated  in  order  to  measure  variations  of 
cost-value,  or  of  esteem-value.  Naturally,  difleront  Icinds  of  value  must  be 
measured  in  different  ways,  and  the  ways  required  for  measuring  variations  of 
other  kinds  of  value  are  not  the  ways  required  for  measuring  variations  of  ex- 
change-value. 


BpyrwEP:x  one  thing  and  all  others  33 

When  A  thus  varies,  say  rising  or  falling  50  per  cent.,  in  re- 
lation to  every  one  of  the  other  things,  all  these  remaining  un- 
changed amongst  themselves,  as  they  must  do  under  the  first 
supposition,  it  is  evident  that  A  has  varied  in  the  same  propor- 
tion, rising  or  falling  oO  per  cent.,  in  exchange-value  in  all 
these  other  things  together.  In  other  words,  a  variation  in  the 
particular  exchange-values  of  any  one  thing  in  the  same  pro- 
portion in  all  other  things  singly  is  a  variation  in  its  general 
exchange-value  in  the  same  proportion  in  all  other  things  col- 
lectively. Or  more  briefly,  lohen  all  the  partieular  exchanye- 
values  of  a  thing  in  other  things  vary  alike,  their  common  vari- 
ation is  the  variation  of  its  general  exchange-value  in  all  other 
things  (Proposition  XVII.). 

It  may  be  noticed  that  absolutely  all  the  particular  exchange- 
values  of  a  thing  cannot  vary ;  for  its  particular  exchange- 
value  in  itself  (or  in  its  mates  within  its  class)  never  varies  (ac- 
cording to  Proposition  II.).  Hence  the  need  of  confining  the 
statement  to  the  particular  exchange-values  in  other  (i.  e.,  in 
other  kinds  of)  things, 

§  2.  But  when  B  and  C  and  D  and  every  other  thing  have, 
say,  fallen  by  33  J  per  cent,  in  exchange-value  in  A,  it  is  equally 
plain  that  B  and  C  and  D  and  every  other  thing  have  not  fallen 
so  much  in  exchange-value  in  all  other  things ;  for  they  have 
not  fallen  relatively  to  one  another.  All  A's  particular  ex- 
change-values in  other  things  have  risen  in  the  same  propor- 
tion, namely  by  50  per  cent.,  and  therefore  A's  general  ex- 
change-value in  all  other  things  has  risen  in  that  proportion. 
But  only  one  particular  exchange-value  of  B,  of  C,  of  D,  etc., 
has  fallen,  while  all  the  other  particular  exchange-values  of  B, 
of  C,  of  D,  etc.,  have  remained  unchanged;  therefore  their 
totals  of  particular  exchange-values,  that  is,  their  general  ex- 
change-values in  all  other  things,  have  been  less  altered.  Of 
course  the  reverse  takes  place  in  the  case  of  a  fall  of  A.  If  A 
falls  in  exchange-value  in  every  other  thing  in  the  same  pro- 
portion, it  falls  in  exchange-value  in  all  other  things  in  that 
proportion ;  and  every  one  of  the  other  things  rises  in  exchange- 
value  in  it  alone  in  inverse  proportion  without  rising  in  any- 
3 


34  THE    CORRELATION    OF    EXCHANGE- VALUES 

thing  else,  and  therefore  every  one  of  tliem  rises  in  exchange- 
value  in  all  other  things  in  a  proportion  smaller  than  the 
inverse.  Thus  we  have  the  general  law :  A  variation  of  any 
one  thing  in  exchange-value  in  the  same  proportion  relatively  to 
all  other  things  involves  an  opposite  variaiion  of  every  one  of  the 
others  in  exchange-value  in  all  other  things  to  a  s:)naller  extent  than 
the  inverse  jyroportion  (Proposition  XVIII.). 

And  evidently,  in  sucJi  cases,  the  extent  of  the  opposite  varia- 
tions will  he  smaller  and  smaller  the  more  numerous  are  the  other 
things  (Proposition  XIX.).  We  perceive  this  by  supposing  at 
first  the  existence  of  only  three  articles  and  then  extending  the 
number.  Let  A,  as  before,  be  the  article  which  rises  in  ex- 
change-value in  every  other  single  thing,  say  by  50  per  cent. 
Then,  as  all  the  other  things  are  in  the  same  box,  having  the 
same  exchange-values,  we  need  to  .follow  the  fortunes  of  only 
one  of  them,  say  B.  Now  with  only  three  articles,^  B  was  at 
first  equivalent  to  A  and  to  C,  and  after  the  rise  of  A  it  is 
equivalent  to  f  A  and  to  C  ;  with  four  articles,  it  Avas  equiva- 
lent to  A,  to  C  and  to  D,  and  later  is  equivalent  to  |A,  to  C 
and  to  D ;  with  five,  it  was  equivalent  to  A,  to  C,  to  D  and  to 
E,  and  later  is  equivalent  to  f  A,  to  C,  to  D  and  to  E ;  and  so 
on.  It  is  plain  that  the  larger  the  number  of  other  things,  the 
larger  is  the  number  of  its  particular  exchange-values  which  B 
retains  unchanged,  and  the  comparatively  smaller  and  less  ap- 
preciable becomes  the  loss  of  its  exchange-value  in  A — or 
smaller  and  less  appreciable  the  loss  of  its  purchasing  power 
over  A  compared  with  its  purciiasing  power  over  the  other 
things  which  remain  unchanged  ; — and  consequently,  the  whole 
l)eing  enlarged,  the  smaller  is  the  fixed  loss  in  one  member 
relatively  to  the  remainder.  With  very  many  articles,  a  small 
change  in  A  relatively  to  the  rest,  may  be  a  practically  inap- 
preciable change  in  B  and  the  others  relatively  to  all  other 
things  as  a  whole.^     But  theoretically,  in  any  such  case,  how- 

1  W^e  must  begin  with  three  in  order  to  get  two  others,  so  as  to  make  the 
plural.  If,  however,  we  started  with  only  two  tilings,  having  only  one  other  for 
"all  the  others,"  the  opposite  variation  would  be  exactly  in  the  inverse  propor- 
tion, this  condition  reducing  to  the  condition  in  Propositions  IX.  and  X.  This  is, 
for  really  other  things,  the  unattainable  limit  of  greatness  in  the  opposite  variation. 

2  Of  course  in  all  this  passage  the  sizes  of  the  other  articles  are  supposed  to  be 


BETWEEN  ONE  THING  AND  ALL  OTHERS        35 

ever  small  the  change  of  A,  there  is  some  change  in  B  and  the 
others  relatively  to  all  other  things  as  a  whole."^  The  reverse 
is  equally  plain  if  we  suppose  A  to  fall. 

§  3.  What  is  said  of  the  correlation  of  exchange- values  in 
the  case  of  things  in  general,  is  of  course  true  when  one  of 
them  is  money.  To  reinterpret  these  relations  in  terms  of  prices, 
is  a  simple  matter.  (1)  If  money  rises  in  exchange-value  in 
every  other  article  by  j)  per  cent.,  it  rises  in  exchange-value  in 
all  other  things  by  p  per  cent.  But  its  rise  in  exchange-value 
in  every  other  article  by  p  per  cent,  is  the  same  as  a  fall  of 

every  one  of  these  in  price  by   per  cent.       Therefore  a 

uniform  fall  of  price  by per  cent,  is  a  rise  of  money  in 

exchange-value  in  all  other  things  by  p  per  cent.,  that  is,  in  in- 

p 
verse  proportion.     And  while  every  article  has  fallen  by  :j 

per  cent,  in  price,  that  is  also,  by per  cent,  in  exchange- 
value  in  money,  every  article  has  fallen  to  a  much  smaller  ex- 
tent in  exchange-value  in  all  other  things.*  And  similarly  in 
the  case  of  a  fall  of  money  or  a  rise  of  prices.  Yet  we  some- 
given.  If  we  increase  their  number  simply  by  breaking  them  up  into  parts,  the 
extent  of  the  opposite  change  in  tlieir  exchange-values  is  not  thereby  affected. 
The  sizes  of  articles,  or  classes,  will  be  treated  of  later. 

'  If  it  be  theoretically  permissible  to  suppose  an  infinite  number  of  equally 
larger  other  things,  then  the  opposite  change,  no  matter  how  much  A  or  any  finite 
number  of  things  finitely  rise  in  evei-y  one  of  the  common  other  things,  the  op- 
posite A'ariation  of  every  one  of  these  in  exchange-value  in  all  other  things  is, 
even  theoretically,  infinitely  small,  or  nil,  that  is,  there  is  no  variation  in  their 
exchange- values  in  all  other  things.  This  is  the  unattainable  limit  of  smallness 
in  the  opposite  variation. 

*  It  is  a  mistake  to  say  that  the  "general  value"  of  these  articles  has  not 
changed,  as  said  by  J.  Prince-Smith,  Valeur  et  monnaie,  Journal  des  Economistes 
Dec.  1853,  p.  373.  But  if  we  purposely  ignore  the  exchange-value  of  money  on 
the  ground  that  money  is  only  an  intermediary  in  exchanges  of  goods,  it  is  true 
to  say  that  the  geueral  exchange-value  of  these  articles  in  all  other,  or  in  all, 
articles  except  money,  has  not  changed.  We  should  remember,  however,  that 
money  is  not  an  intermediary  merely  in  contemporaneous  exchanges,  but  that 
frequently  a  long  interval  i>asses  between  the  time  when  we  get  money  for  our 
property  and  the  time  when  we  get  other  property  for  our  money,  so  that  the  in- 
tervening state  of  the  exchange-value  of  money  is  very  important.  Still  the  im- 
portance seems  to  be  sui  generis. 


36  THE    CORRELATIOX    OF    EXCHANGE- VALUES 

times  hear  such  faulty  expressions  as  this  :  If,  prices  rising  by 
50  per  cent.,  money  depreciates  by  33^  per  cent.,  commodities 
have  appreciated  by  50  per  cent.;  or  conversely.  It  is  true 
that  commodities,  each  singly,  have  appreciated  by  50  per  cent, 
in  money  ;  but  to  say  simply  that  commodities  have  appreciated 
50  per  cent,  means  that  they  have  risen  50  per  cent,  in  ex- 
change-value simply,  which  is  not  true.^  Again  (2)  if  money 
rises  by  p  per  cent,  in  exchange-value  in  one  article  alone 
while  retaining  unaltered  its  exchange-value  in  every  other 
article,  this  is  only  another   way  of  saying  that   the   price  of 

that  one  article  has  fallen  by    ^  per  cent.,  while  the  prices 

of  all  other  articles  has  remained  unchanged.  Then  the  ex- 
change-value of  that  one  article  in  every  other  thing,  and  in 

all  other  things  has  fallen  by  z per  cent.;  and  the  exchange- 
value  of  every  other  article,  as  of  money,  in  that  one  thing, 
has  risen  hy  p  per  cent.,  but  the  exchange-value  of  money,  and 
of  every  other  article,  in  all  other  things,  though  it  has  risen 
somewhat,^  has  by  no  means  risen  to  that  extent. 

III. 

§  1.  A  corollary  from  the  preceding  laws  is  this  :  There 
cannot  he  a  rariafio)i  in  the  exchange-value  of  one  thing  alone 
(Proposition  XX.). ^  There  can,  of  course,  be  a  rise  or  a  fall 
in  exchange-value  of  one  thing  alone,  since  one  thing  may  rise 
without  anything  else  rising,  and  one  thing  may  fall  without 
anything  else  falling.  But  the  moment  one  thing  alone  so  rises 
or  falls  in  exchange-value,  all  the  others  inversely  fall  or  rise. 

^  Another  similarly  faulty  expression  is  tli;it  in  the  case  of  a  rise  of  prii-es 
"values"  have  risen;  and  conversely.  It  is  only  values  in  money,  or  values 
expressed  in  money,  that  have  changed  exactly  as  the  prices.  Of  course  on  the 
stock-market,  where  the  only  values  thought  of  are  money-values,  this  expression 
is  natural. 

"  As  correctly  said  by  Prince-Smith,  ojy.  cif.,  p.  373,  and  by  .levons,  B.  15,  p. 
20. 

^  So  Walras,  Elements,  1st  ed.,  pp.  167-1(58,  2d  ed.,  p.  47)6.  This  is  very  differ- 
ent from  what  is  referred  to  in  Note  1  in  Sect.  I.  of  this  chapter.  Here  he  is 
speaking  of  exchange- value  ;  there,  what  he  had  in  mind  was  estoom-value. 


IN\'ARIABILITY    OF    OXE    THING    ALONE  37 

Hence  it  is  a  solecism  to  speak  simply  of  a  change  of  one  thing 
alone  in  exchange-value.  There  may  be  a  change  of  one  price 
alone  ;  but  as  we  have  just  seen,  this  is  not  a  change  in  the  ex- 
change-value of  the  one  article  alone.  Furthermore  there  may 
be  a  change  in  the  use-value,  in  the  cost-value,  and  perhaps  even 
in  the  esteem-value,  of  one  thing  alone.  But  here  is  only  one 
more  reason  for  distinguishing  use-value  and  cost-value  and 
esteem-value  from  exchange-value.  Likewise  it  is-  conceivable 
that  a  cause  may  act  upon  one  thing  alone  to  make  its  price 
change,  or  its  use-value  change,  or  its  cost-value  or  esteem-value 
change,  without  affecting  the  price,  use-value,  cost-value  or  per- 
haps (although  this  is  questionable)  even  the  esteem-value,  of 
anything  else.  It  is  not  possible,  however,  for  a  cause  to  act 
upon  one  thing  alone  so  as  to  render  it  of  greater  or  smaller  ex- 
change-value, since  this  cause  will  also  act  upon  all  other  things 
to  render  them  of  smaller  or  greater  exchange-value  in  this  thing 
and  thereby,  to  some  extent,  in  all  things.^ 

§  2.  What  is  here  said  shows  the  wrongness  of  a  doctrine 
sometimes  taught  concerning  "value,"  supposedly  in  the  sense 
of  exchange-value.  This  is  that  when  a  sensible  rise  or  fall 
takes  place  in  the  value  or  price  of  one  thing  alone  (or  in  the 
value  of  money,  and  therefore  in  all  prices  in  the  opposite  di- 
rection), the  rest  remaining  tolerably  steady  relatively  to  one 
another,  there  is  reason  for  holding  that  the  change  is  altogether 
in  the  one  thing  and  not  at  all  in  the  others — that  is,  that  the 
one  thing  alone  has  changed  in  "  value,"  and  the  others  have 
remained  unchanged  in  "  value."  The  reason  assigned  is  that 
it  is  easier  to  explain  these  phenomena  by  supposing  one  cause 
aifecting  this  one  thing  than  by  supposing  causes  affecting  the 
other  things,  which  causes  would  have  to  be  as  numerous  as  the 
other  things  ;  or,  in  other  words,  that  the  hypothesis  invoking 

^  ilence  there  is  impossibility  iu  the  assumption  made  by  H.  Fawcett  (follow- 
ing J.  S.  Mill,  op.  cit.,  Vol.  I.,  pp.  539-540)  in  his  3Ianual  of  political  economy, 
18fi3,  6th  ed.,  1883,  p.  314,  at  the  commencement  of  his  inquiry  into  "  the  causes 
which  regulate  the  price  of  commodities,"  namely  that  a  variation  of  price  "  is 
always  supposed  to  be  produced  by  something  which  affects  the  value  of  the  com- 
modity and  not  the  value  of  precious  metals";  for  by  "value"  lie  meant  ex- 
change-value. Fawcett  has  been  followed  by  A.  S.  Bolles,  Chapters  in  political 
economy,  1874,  p.  56. 


38       THE  CORRKLATIOX  OF  EXCHANGE- VALUES 

one  cause  is  far  the  simpler  and  more  probable — -just  as  the 
Copernican  theory  is  simpler  and  more  probable  than  the  Ptol- 
emaic.^ 

The  statement  made  in  this  reason  assigned  for  the  doctrine 
in  question  is  in  itself  not  incorrect.  In  default  of  data  for 
finding  the  cause  or  causes  directly,  or  in  our  impatience  at  the 
delay  required  for  such  a  direct  investigation,  it  is  easier  for 
us  to  suppose,  and  we  are  inclined  to  believe,  that  there  is  only 
one  cause  affecting,  or,  as  is  sometimes  carelessly  said,  residing 
in,  the  one  thing  whose  exchange-value  in  all  others  is  most 
prominently  changing.^  But  to  take  this  view  of  the  matter  as 
a  reason  for  supposing  a  change  in  exchange-value  only  in  the 
one  thing,  is  absolutely  incorrect.  We  can — it  being  supposed 
all  along  that  we  know  the  particular  variations — calculate  with 
mathematical  precision,  when  we  have  fully  developed  the 
theory  of  exchange-value,  almost  the  exact  variation  in  the  ex- 
change-value (in  all  other  things,  and  also  in  all  things)  of  the 
one  thing  and  almost  the  exact  variations  in  the  (similar)  ex- 
change-values of  all  the  other  things.  There  is  no  occasion  for 
employing  the  law  of  probabilities  at  all. 

The  explanation  of  this  doctrine  itself — and  of  the  use  of  the 
law  of  probabilities  with  regard  to  the  causes  as  a  reason  for 
applying  the  same  to  the  changes  of  "value" — is  the  retention 
in  the  idea  of  "v^alue"  of  the  ideas  of  cost-value  and  of 
esteem-value.  In  the  case  of  cost-value,  and  perhaps  also  of 
esteem-value,  it  would  be  possible  for  the  change  to  reside  only 
in  the  cost-value,  or  in  the  esteem-value,  of  the  one  thing  ; 
and  accordingly  here,  in  defiiult  of  knowledge  to  the  contrary, 
it  is  more  probable,  it  is  a  more  likely  hypothesis,  that  the 
change  has  taken  place  in  the  cost-value,  or  in  the  esteem-value, 
of  the  one  thing  alone.  This  is  because,  at  least  in  the  case 
of  cost-value,  this  value  is  wholly  dependent  upon  the  labor 
required  for  producing  the  thing,  and  so  is  measured  by  that 

^  Rieardo,  Works,  p.  13 ;  A.  rournot,  Jircherches  snr  les  principes  mathc- 
matiques  de  la  thmrif  de  la  richesse,  1838,  pp.  18-11* ;  the  latter  followed  (with 
addition  of  the  simile)  by  C.  Gide,  Principes  d'economie  politique,  1884,  pp. 
81-83. 

*  Thus  this  hypothesis  in  regard  to  the  cunse,  liiit  not  in  regard  to  tiie  cliange 
in  value,  was  used  l)y  Jevons,  1$.  22,  pp.  l!»-2(l,  15.  LM,  p.  ir)(). 


INVARIABILITY    OF    ONK    THING    ALONE  39" 

labor,  that  is,  is  measured  by  a  measurement  of  its  cause.  Now 
if  it  should  really  happen  that  oue  thing  alone  rose  or  fell  in 
cost-value,  the  relation  of  its  cost-value  to  the  cost-value  of  all 
the  other  things  would  have  varied,  and,  if  it  rose  compared 
with  them,  they  would  have  fallen  compared  with  it ;  and  if, 
as  is  sometimes  the  case,  the  exchange-value  of  the  thing  varies 
with  its  cost-value,  its  exchange- value  would  rise  in  other 
things  and  theirs  would  fall  in  it  (and  consequently  in  exchange- 
value  simply).  Here  then  we  should  be  reaching  a  right  con- 
clusion by  means  of  the  law  of  probabilities  if  we  supposed  the 
change  of  cost-value  to  be  solely  in  the  one  thing  ;  but  we 
should  be  violating  the  assumed  conditions  if  we  concluded 
either  that  the  relative  cost-value  of  the  thing  compared  with 
the  others  has  alone  changed  and  not  also  their  relative  cost- 
values  compared  with  its,  or  that  the  exchange-value  of  the  one 
thing  (meaning  its  exchange-value  simply,  /.  e.,  in  all  other 
things,  or  in  all  things,  or  compared  with  the  exchange-values 
of  the  other  things)  has  alone  changed  and  not  also  the  ex- 
change-values of  the  other  things  (meaning  their  exchange- 
values  simply,  /.  c,  in  this  thing  and  in  all  other  or  in  aU 
things,  or  compared  with  this  thing's  exchange-value).  The 
very  data  assumed  require  these  correlative  changes  both  in  re- 
lative cost-values  and  in  exchange-values — for  all  exchange- 
value  is  relative.^ 

IV. 

§  1.  Another  corollary   from   the    above-given    laws  is  the 

already  mentioned  distinction    between  exchange-value  in  all 

^  There  is,  therefore,  no  need  of  speaking  of  "  rehxtive  exchange-value."  In 
fact,  the  very  term  "exchange"  in  this  compound  term  has  the  meaning  of 
"  relative,"  so  that  "  exchange- value  "  is  similar  to  "  relative  value."  Sometimes 
exchange-value  is  relative  cost-value  ;  always  it  is  relative  esteem-value.  In  the 
course  of  time  a  thing  retains  the  same  exchange- value  if  it  retains  the  same  rela- 
tive esteem-value,  that  is,  whether  its  esteem-value  vary  or  not,  if  it  varies  or  not 
just  as  the  esteem-values  of  all  other  things  on  the  average  vary  or  not.  Or  in 
the  course  of  time  its  exchange-value  varies  if  its  relative  esteem-value  varies, 
that  is,  if  its  esteem-value  varies  compared  with  the  average  of  the  esteem-values 
of  all  other  things,  for  instance  rising  (1)  if  it  rises  more  than  they  do,  or  (2)  if 
it  rises  while  they  are  stationary,  or  (3)  if  it  rises  while  they  fall,  or  (4)  if  it  is 
stationary  while  they  fall,  or  (5)  if  it  falls  less  then  they  do— in  the  same  five 
typical  relations  we  have  already  noticed  in  another  connection. 


40  THE    CORRELATION    OF    EXCH  AN<;  K-VA  U' EH 

other  things  and  exchange-value  in  all  triings.  For  the  law 
that  if  one  thing  alone  rises  in  exchange-value  in  every  other 
thing  to  the  same  extent,  it  rises  exactly  to  that  extent  in 
exchange-value  in  all  other  things,  is  perfectly  evident.  And 
also  the  law  is  equally  plain  that  every  one  of  those  other 
things  has  sunk  somewhat  in  exchange-value  in  all  other 
things.  Putting  these  two  laws  together,  we  see  that  when  a 
thing  rises  a  certain  percentage  in  exchange-value  in  all  other 
things,  as  it  rises  to  that  extent  in  things  that  have  themselves 
sunk  somewhat  in  exchange-value  in  all  other  things,  it  has 
not  risen  quite  that  percentage  in  exchange-value  after  all,  but 
to  a  somewhat  lesser  extent.  The  performance  is  comparable 
with  the  rise  of  a  body  above  a  plane  which  is  depressed  by 
the  force  raising  the  body  ;  in  which  case  the  measurement  of 
the  rise  of  the  body  in  relation  to  the  plane  gives  a  greater  re- 
sult than  the  true  rise.  Similarly  in  the  reverse  case  of  the  fall 
of  a  body  by  which  the  plane  from  which  it  fiills  is  repelled 
upward,  the  measurement  of  the  fall  of  the  body  from  the  plane 
also  gives  too  great  a  fall.  So  a  fall  of  one  thing  in  exchange- 
value  in  all  other  things,  beinsi;  a  fall  which  raises  their  ex- 
change-values  somewhat,  is  not  so  great  a  fall  really  as  it  ap- 
pears to  be,  judged  by  comparison  with  them  only. 

The  explanation  of  the  difference  is  obvious.  In  treating 
of  the  exchange-value  of  one  thing  in  all  other  things  we  are 
using  all  other  things  as  the  standard  ;  but  then  in  treating  of 
the  exchanse-value  of  anv  one  of  the  other  thino;s  also  in  all 
other  things  we  are  using  a  different  standard,  for  this  standard 
excludes  the  latter  thing  which  was  included  in  the  former 
standard,  and  includes  the  former  thing,  which  was  excluded 
from  that  standard.  These  two  standards,  then,  are  shifting 
standards  ;  but  together  they  include  all  things.  Thus  when 
we  perceive  that  a  thing  which  has  risen  a  certain  extent  in 
exchange-value  in  all  other  things  has  not  risen  quite  so  high 
in  exchange-value,  by  this  exchange-value  we  mean  an  ex- 
change-value measured  in  a  standard  in  which  not  only  all  the 
other  things  are  included,  but  also  the  thing  itself  of  which  the 
altered  exchange- value  is  being  measured.     Thus  this  exchange- 


TWO    KIXDS    OF    GEXEPvAL    P:XCHAXGE-VALUE  41 

value  is  exchange-value,  not  in  all  other  things,  but  in  all 
things.  And  it  is  the  superior  and  final  exchange-value,  the 
one  more  than  any  other  appearing  to  deserve  the  single  term 
exchange-value.  We  might  even  be  tempted  to  call  it  "  abso- 
lute exchange-value,"  except  that  there  are  metaphysical  and 
logical  objections  to  the  use  of  this  term.  These  objections 
could  not  hold  against  calling  it  "  universal  exchange-value." 
But  perhaps  it  is  as  well  not  to  give  it  any  name. 

The  simple  term,  then,  exchange-v^alue,  or  general  exchange- 
value,  is  slightly  ambiguous.  The  same  ambiguity  extends  to 
the  related  terms,  appreciation  and  depreciation,  and  the  like. 
Thus  a  thing  which  appreciates  a  certain  percentage  in  ex- 
change-value in  all  other  things,  does  not  appreciate  quite  so 
much  in  exchange-value  in  all  things.  Yet  almost  all  writers 
on  the  subject  have  measured  appreciation  and  depreciation,  or 
in  general  any  variation  in  exchange-value,  in  the  former  way 
only.  Thus  it  is  commonly  asserted  that  if  all  other  things 
remain  unchanged  amongst  themselves,  including  money,  the 
change  of  one  thing  relatively  to  any  one  of  them,  and  so  its 
change  in  price,  exactly  measures  its  change  in  general  ex- 
change-value ;  ^  or  again  that  if  all  prices  change  alike  (or  on 
an  average)  to  a  certain  extent,  money  has  changed  to  the  in- 
verse extent  in  exchange-value  simply.^  Such  statements  are 
only  half  true.  Tliey  are  true  of  the  variations  in  exchange- 
value  in  all  otiier  things ;  but  they  are  not  true  of  the  varia- 
tions in  exchange-value  in  all  things. 

§  2.  The  differences  between  these  two  kinds  of  general  ex- 
change-value and  the  laws  common  to  them  both,  may  be  briefly 
surveyed.  It  is  evident  that  a  thing's  exchange-value  in  all 
tilings  always  varies  less  than  its  exchange-value  in  all  other  things 
(Proposition  XXI.).     For,  given  the  deviation  of  a  thing's  ex- 

1  JE.  g.,  J.  S.  Mill,  op.  cit.,  Vol.  I.,  pp.  539-540. 

-So  Prince-Smith,  op.  cit.,  p.  373.  J.  S.  Mill  says  that,  caeteribus  paribus, 
prices  vary  directly,  the  exchange-value  of  money  inversely,  with  the  quantity  of 
money,  op.  cit.,  Vol.  II.,  pp.  16-17.  It  is  usual,  after  getting  a  method  or  formula 
for  measuring  the  variation  of  the  average  of  prices,  simply  to  invert  this  as  the 
measure  of  appreciation  or  depreciation  of  money  ;  so  Levasseur,  B.  18,  p.  195; 
Jevons,  B.  22,  pp.  53-54,  58  ;  Drobisch,  B.  29,  p.  39,  B.  30,  p.  149  ;  Lehr,  B.  m,  p. 
40;  Nicholson,  B.  94,  pp.  317-318;  Lindsay,  B.  114,  p.  2(3. 


42       THE  COKRELATIOX  OF  EXCHAXGP:- VALUES 

change- value  in  all  other  things,  as  its  exchange- value  in  itself 
never  varies,  when  this  unchanged  particular  exchange-value  of 
itself  in  itself  is  added  to  the  standard  by  which  its  deviation  is 
measured,  it  must  have  the  effect  of  lessening  the  deviation 
somewhat.  But  it  cannot  annul  the  deviation,  since  there  is 
some  jmrticular  variation  as  before,  which  must  still  be  counted. 
Therefore  ilte  one  yvneral  exchange-value  of  a  thine/  cannot  vary 
loithout  the  other  varylny  also  (Proposition  XXII.).  And,  of 
course,  the  lessening  just  shown  cannot  deflect  the  deviation  in 
the  opposite  direction  ;  wherefore  the  two  kinfls  of  general  ex~ 
ichange-valite  always  deviate  from  constancy  in  the  same  direction 
(Proposition  XXIII.).  Further,  as  we  have  seen  that  when 
one  thing  alone  rises  in  exchange-value  in  all  other  things,  the 
others  fall  somewhat,  and  less  the  more  numerous  they  are,  and 
conversely,  so  we  now  see  that  there  are  two  rises  of  the  one 
thing  and  consequently  also  two  falls  of  the  other  things,  and 
conversely.  It  is  now  plain  that  in  both  cases,  the  divergence 
between  the  two  deviations  of  the  two  ki)ids  of  general  excJiange- 
v(due  is  smaller  fJte  more  numerous  are  the  articles  (Proposition 
XXIV.).  For  the  divergence  is  determined  by  the  difference 
between  the  whole  number  of  things  minus  one  thing  and  the 
whole  number  of  things  compared  with  the  whole  number  of 
things,  and  this  comparative  difference  is  smaller  the  larger  the 
whole  number  of  things.  Thus,  for  example,  if  we  know  the 
variation  of  a  thing  in  exchange-value  in  all  otJier  things  (as 
when  conditions  satisfy  Proposition  XVII.),  then  we  know 
that  its  variation  in  exchange-value  in  all  things  is  almost  as 
great,  if  the  other  things  are  very  numerous  (or  very  much 
larger  than  the  one  thing,  or  than  its  class).  Similarly,  given, 
the  number  of  (^equal)  articles,  the  divergence  between  the  two  devi- 
ations is  always  in  the  same  proportion  (Proposition  XXV.). 
For  here  the  comparative  difference  is  the  same  in  all  the 
measurements.  We  shall  later  find  formulae  from  which,  in 
some  cases,  the  exact  proportions  can  be  derived,  if  desired^ 
and  a  very  simple  proposition  expnjssing  them. 

§  3.    Upon  these  laws  of  variation  and   divergence  follows  a 
law  of  constancy  and   identity.      It  is  plain  that  if  there  is  no 


TWO    KINDS    OF    (rENERAL    EXCHANGE-VALUE  43 

variation  of  a  thing's  exchange-value  in  all  other  things,  there 
can  be  no  variation  of  the  thing's  exchange-value  in  all  things, 
since  nothing  is  added  in  the  latter  case  to  make  a  variation  not 
existing  in  the  former.  Nor,  reversely,  if  there  is  no  variation 
of  a  thing's  exchange-value  in  all  things,  can  there  be  a  vari- 
ation of  the  thing's  exchange-value  in  all  other  things,  because, 
if  an  element  of  variation  existed  in  the  latter  case,  it  would 
have  existed  in  the  former  also.  In  other  words,  constanrt/  of 
the  one  kind  of  general  exchange-vahie  pomessed  by  anything  is 
constancy  of  the  other  (Proposition  XXVI.).  That  is,  again, 
the  two  kinds  of  general  exchange- value  coincide  and  are  identical 
when  either  of  them  is  constant.  Another  demonstration  of  this 
theorem  may  be  offered  as  follows  :  If  changes  in  a  thing's 
particular  exchange-values  occur  so  that  its  exchange-value  in 
all  other  things  rises,  its  exchange-value  in  all  things  rises,  but 
less  ;  and  if  such  changes  occur  so  that  its  exchange-value  in 
all  other  things  falls,  its  exchange-value  in  all  things  falls,  but 
less  ;  therefore  if  we  suppose  these  changes  on  each  side  to  be 
smaller  and  smaller  until  the  change  in  the  thing's  exchange- 
value  in  all  other  tilings  is  so  small  as  to  be  inappreciable,  its 
exchange-value  in  all  things,  always  being  even  smaller,  must 
be  still  less  appreciable,  and  when  the  former  becomes  zero,  the 
latter  must  also  be  zero. 

On  account  of  this  coincidence  of  the  two  kinds  of  general 
exchange- value  at  rest,  we  are  justified  in  subsuming  them  under 
the  same  term  ;  and  whenever  we  have  occasion  to  speak  of 
constant  general  exchange-value,  or  simply  of  constant  exchange- 
value,  there  is  no  need  of  distinguishing  between  the  two  kinds, 
which  separately  exist  only  when  there  is  variation.^ 

'  Beginning  at  a  given  period,  we  see  that  if  there  is  a  variation  of  a  tiling's 
general  exchange-value,  this  splits  up  into  two  variations  from  the  original  co- 
incident positions  of  the  two  kinds  of  general  exchange-value  ;  but  if  there  had 
been  constancy  of  the  thing's  genei'al  exchange-value,  there  would  have  been  no 
such  splitting,  and  the  two  kinds  would  not  have  manifested  their  distinctness. 
Now  when  such  a  variation  and  such  a  splitting  have  occurred,  if  we  start  over 
again  from  the  second  period  as  a  new  starting  point,  the  same  phenomena  may 
repeat  themselves;  that  is,  if  from  this  starting  point  there  is  constancy,  the 
general  exchange-value  remains  undivided,  but  if  there  is  variation,  it  divides 
again  into  two.  And  so  on  from  any  later  period  as  a  new  starting  point.  In 
every  case  the  two  kinds  of  general  e.xchange-value  at  the  second  period,  com- 


44  THE    CORRELATION    OF    EXGHAXdE-VALUES 

V. 

§1.  Constancy  of  general  exchange- value  may  happen  under 
two  different  conditions  of  things.  The  first  is  the  simplest 
and  most  obvious.  It  is  evident  that  if  all  the  particular  ex- 
change-values of  a  thing  in  all  other  things  remain  constant,  its 
general  exchange-value  in  all  other  things  and,  since  its  ex- 
change-value in  itself  must  remain  constant,  its  general  ex- 
change-value in  all  things  remain  constant.  This  condition 
takes  place  when  there  are  no  changes  whatsoever  between  the 
particular  exchange-values  of  any  thing  in  any  other  amongst 
all  things  ;  so  that  the  general  exchange-value  not  only  of  the 
thing  in  question,  but  of  everything  else,  is  without  change. 
Thus,  if  all  tilings  retain  the  same  exchange-values  relatively  to 
one  another,  the  general  excJiange-value  of  any  one  thing  is  con- 
stant (Proposition  XXVII.). 

The  second  condition  is  not  quite  so  obvious,  but  is  equally 
demonstrable.  For  evidently  it  is  possible,  when  one  thing 
changes  alike  in  exchange-value  in  all  others  except  one,  for  all 
the  rest  beside  these  two  to  remain  unchanged  in  exchange- 
value  in  all  other  things,  and  consequently  in  all  things,  pro- 
vided that  the  other  one  thing  change  in  respect  to  them  in  a 
manner  to  compensate  for  the  change  of  the  first  one.  And 
consequently,  any  two  compensating  for  each  other,  or  several 
compensating  for  one  or  for  several,  or  many  for  many,  it  is 
possible  for  one  thing  to  retain  the  same  general  exchange- value, 
in  spite  of  changes  in  any  two  or  in  any  higher  number  of  others 
up  to  all  others. 

§  2.  Here  we  meet  with  an  error  which  would  be  fatal  in 
our  subject.  A  few  economists  have  asserted  that  the  general 
exchange-value  of  anything  can  remain  unchanged  only  under 
the  first  condition,  and  as  that  condition  is  never  fulfilled  in 
the  actual  world  except  over  the  briefest  of  intervals,  they 
have  denied  even  the  possihilify  of  any  one  thing,  or  in  especial, 
money,  retaining  the  same  general  exchange-value  through  the 

pared  with  the  first,  may  have  remained  the  same  or  have  divided  ;  but  in  every 
case  the  two  kinds  at  tiie  first  period,  to  be  compared  with  tlie  set-ond,  are  taken 
as  identical. 


THE    POSSIBILITY    OF    CONSTANCY  45 

course  of  time^ — and  one  writer  has  even  rejected  the  expression 
"  stable  value  "  as  a  collocation  of  words  without  meaning,  like 
"  triangular  square."  ^  The  same  position  has  been  taken  by 
Roscher  concerning  general  purchasing  power  ^ — with  better 
reason,  as  we  shall  see,  for  one  sense  in  which  this  term  is 
used.  Now  if  this  view  were  correct  in  regard  to  exchange- 
value,  that  is,  if  the  constancy  of  general  exchange-value  under 
any  changes  of  particular  exchange-values  is  even  theoretically 
impossible,  or  inconceivable,  it  would  be  absurd  to  speak  of 
a  greater  or  less  variation  of  a  thing's  general  exchange- 
value  through  the  course  of  time,  since  this  expression  always 
has  reference  to  its  variation  from  a  supposed  constant  position, 
and  if  this  has  no  possible  existence,  the  magnitude  of  the  vari- 
ation has  no  possible  existence — nay,  the  variation  itself  could 
not  possibly  exist,  and  the  general  exchange-value  of  a  thing 
at  one  time  would  be  wholly  incommensurable  with  its  general 
exchange- value  at  another  time,  as  indeed  has  been  maintained 
by  some  extremists.^  Then  not  only  to  measure  a  thing's  con- 
stancy in  general  exchange-value  would  l)e  an  idle  endeavor, 
but  also  to  measure  its  variation  in  general  exchange- value 
would  be  meaningless.  And  the  statement  frequently  made 
by  the  very  persons  who  deny  the  possibility  of  constancy  of 
general  exchange- value,  or  the  possibility  of  measuring  it,  that 
the  desirable  feature  in  money  is  that  it  should  vary  as  little  as 
possible  in  exchange-value,  would  be  an  absurdity."  * 

^  Galiani,  Delia  moneta,  1750,  ed.,  Custodi,  Vol.  II.,  p.  11 ;  :N[cCulloch,  Polit- 
ical economy,  p.  213  (Note  to  Wealth  of  nations,  p.  439) ;  Prince-Smitli,  op.  cit.,  p. 
373;  Macleod,  Theonj  of  banking.  Vol.  I.,  p.  69,  Thcorij  of  credit,  Vol.  I.,  p.  176; 
Walras,  Elements,  1st  ed.,  pp.  167-168,  185,  2d  ed.,  p.  4o6;  ^lartello,  op.  cit.,  pp. 
402-403. — Rossi  seems  to  have  had  the  same  idea  when  he  said  there  can  be  no 
measure  of  value  because  as  a  standard  it  should  be  invariable  and  as  a  value  it 
must  be  variable,  Cours  d'economic  politique,  1840,  Vol.  I.,  p.  1.30. 

^  F.  Ferrara,  Introduction  to  !Martello's  work,  p.  CXLIX. 

3  B.  32,  ?  127. 

'^  E.  g.,  A.  Held  maintained  that  we  cannot  properly  say  that  in  a  financial 
crisis  the  exchange-value  of  money  is  greater  than  it  was,  but  only  greater  than  it 
would  have  been  had  no  crisis  arisen,  iVoc/;  eimmal  liber  den  Preis  des  Geldes. 
Ein  Beitrag  zur  Milnzfrage,  Jahrbiicher  fiir  Nationaloekonomie  und  Statis- 
tik,  .lena  1871,  p.  328.  He  might  as  well  say  that  when  a  yard-stick  expands 
under  the  influence  of  heat,  it  is  not  longer  than  it  was,  but  only  longer  than  it 
would  have  been  had  it  not  been  heated.  How  it  is  we  can  compare  a  value  with 
what  it  would  have  been,  but  not  with  what  it  was,  is  not  explained. 

5  Among  the  writers  mentioned  in  Notes  1  and  2  only  Ferrara  and   Martello 


46  THE    CORRELATIOX    OF    EXCHANGE- VALUES 

§  3.  But  the  reasoning  by  which  is  maintained  the  denial  of 
the  possibility  of  a  constant  exchange- value  through  variations 
of  particular  exchange-values,  is  the  plainest  of  fallacies.  It  is 
based  on  our  Proposition  XX.  The  exchange-value  of  a  thing 
is  affected  and  altered  if  any  o»c  other  thing  varies  in  exchange- 
value.  Therefore,  it  is  concluded,  its  exchange-value  must  be 
still  more  affected  and  altei*ed,  if  two  other  things,  if  three,  if 
many,  vary  in  exchange-value — that  is,  if  any  other  thing  or 
things  vary  in  exchange-value.  Xow  it  is  plain  that  the  greater 
the  number  of  other  things  that  vary  in  exchange-value,  the 
more  the  exchange- value  of  the  thing  in  question  is  affected; 
but  as  soon  as  it  is  affected  by  two  or  more  other  influences,  the 
possibility  arises  that  it  may  be  affected  in  opposite  directions, 
so  that  the  alteration,  of  its  general  exchange-value,  instead  of 
being  increased,  may  be  diminished,  and  furthermore,  that  it 
may  be  reduced  to  zero  by  the  opposite  influences  exactly 
counterbalancing  each  other. 

This  may  be  rigorously  demonstrated  as  follows.  Suppose 
at  first 

M  =  A=o=B^C^ , 

and  later  A  rises  in  exchange-value  in  all  the  others,  so  that 
M^^  A  =  B^C  ^  ; 

X 

tlicn  M  has  fallen  somewhat  in  general  exchange-value.  Again, 
instead  of  this  rise  of  A,  suppose  B  falls,  so  that 

M  =  A  c  2/B  =  C  =  , 

(in  each  of  these  expressions  x  and  y  being  greater  than  unity); 
then  M  has  risen  somewhat  in  general  exchange-value.  Now 
it  is  evident  that  if  these  two  changes  take  place  together,  so  that 

M^-A^2/BoC=e= , 


X 

are  guiltless  of  tliis  inconsistency.  Walras  tries  not  only  to  measure  general  ex- 
change-value, but  also  to  regulate  the  general  exchange-value  of  money  with  a 
view  to  its  constancy.  Several  other  economists,  who  have,  for  various  reasons, 
denied  that  we  can  measure  exchange-value,  or  that  money  is  a  measure  of  it, 
have  fallen  into  similar  inconsistencj'. 


THE    POSSIBILITY    OF    CONSTANCY  47 

then  the  general  exchange-value  of  M  is  influenced  to  a  fall  by 
the  one  change  and  by  the  other  to  a  rise,  and  it  can  be  'that, 
the  amount  of  the  first  change  being  fixed,  the  influence  of  the 
second  change  may  outweigh  its  influence,  conducing  to  a  fall, 
or  the  influence  of  the  second  change  may  be  insufficient,  per- 
mitting a  rise.  It  is  also  evident,  in  accordance  with  the  prin- 
ciple of  continuity,  that  between  these  higher  and  lower  influ- 
ences of  the  variable  second  change,  compared  with  the  fixed 
influence  of  the  first,  there  must  be  a  point  at  which  that  in- 
fluence exactly  counterbalances  this  influence,  so  that,  the  two 
influences  neutralizing  each  other,  or  having  the  same  effect  as 
if  neither  existed,  the  general  exchange-value  of  M  remains 
unchanged — just  as  two  opposite  and  equal  forces  will  keep  a 
body  at  rest. 

Now  the  rise  of  A  in  exchange- value  in  all  other  things  is 
the  inverse  fall  of  M's  particular  exchange-value  in  A,  and  the 
fall  of  B  in  exchange-value  in  all  other  things  is  the  inverse 
rise  of  M's  particular  exchange-value  in  B.  Therefore,  tlie 
general  exchange-value  of  anything  may  remain  unchanged  in 
spite  of  a  variation  in  one  of  its  particular  exchange-values  (or  an 
inverted  variation  in  the  general  exchange-value  of  one  other  thing), 
if  this  be  compensated  by  an  opposite  variation  in  another  of  its 
particular  exchange-values  {or  an  opposite  variation  in  the  general 
exchange-value  (f  another  thing)  (Proposition  XXVIII.). 

Furthermore,  it  is  obviously  not  necessary  for  the  counter- 
balancing to  be  done  by  a  variation  of  one  other  thing  alone. 
The  compensation  for  the  variation  of  one  thing  may  be  effected  by 
variations,  in  the  opposite  direction,  of  many  others,  or  even  of  all 
the  others  (Proposition  XXIX.).  If  done  by  more  than  one 
other,  it  is,  however,  plain  that  the  variations  of  the  others  in- 
dividually must  be  smaller  than  the  variation  of  only  one  other, 
because  their  combined  influence  constitutes  the  compensation. 
If,  for  instance,  the  rise  of  A  were  already  compensated  by  the 
fall  of  B,  then  a  fall  of  C  would  disturb  the  balance  and  itself 
require  to  be  compensated  either  by  the  rise  of  something  else  or 
by  a  still  higher  rise  of  A.  Thus  a  great  rise  of  A  may  be 
compensated  by  two  lesser  falls,  equal  or  unequal  to  each  other, 


48  THE    CORRELATIOX    OF    EXCHANGE- VALUES 

of  B  and  C  ;  coDsequently,  in  general,  any  rise  of  one  thing,  by 
any  number  of  falls  of  other  things,  and  conversely.  And,  as 
the  compensating  influence  must  be  distributed  over  all  the 
other  changes,  it  is  plain  that,  the  sizes  of  the  things  (or  classes) 
being  given,  the  greater  the  number  of  the  compensating  variations 
the  smaller  must  they  he  i iidividaall y  or  on  an  average  (Proposi- 
tion XXX.), 

Also  it  is  plain  that,  a  certain  variation,  in  one  direction  being 
compensated  by  one  or  many  variations  in  the  opposite  direction , 
it  is  p)Ossible  for  still  another  variation  in  either  direction  to  be 
compensated  by  still  another  or  other  variations  in  the  direction 
opposite  to  it,  and  so  on  indefinitely,  all  at  the  same  time,  the  one 
thing  in  question  remaining  unchanged  in  general  exchange- 
value  so  long  as  all  variations  in  one  direction  are  compensated, 
in  pairs,  or  in  combination,  by  variations  in  the  opposite  direc- 
tion (Proposition  XXXI.). 

§  4.  Now  when  the  one  thing  we  have  in  mind — M  in  the 
above  example — remains  constant  in  general  exchange-value 
under  changes  in  two  or  more  other  things,  it  is  plain  that  all 
the  other  things  that  have  not  varied  relatively  to  it — the  C, 
etc.,  in  the  above  example — have  also  remained  constant  in 
general  exchange-value  (in  accordance  with  Proposition  XIV.). 
Consequently,  the  constancy  of  any  one  thing  in  general  exchange- 
value  is  unaffeeted.  by  the  number  of  other  things  likewise  constant 
in  general  exchange-value,  and  in  any  calculation  it  ought  to  be 
indifferent  whether  we  include  them  or  not  (Proposition 
XXXII. ).  Evidently  it  is  only  things  that  vary  in  general 
exchange-value  that  can  exert  influence  to  cause  a  variation  in 
the  general  exchange-value  of  anything  that  would  otherwise 
be  constant." 

But  with  regard  to  variations  a  distinction  is  to  be  drawn. 
The  things  which  have  varied  in  certain  proportions  relatively  to 
the  thing  or  things  remaining  constant  have  varied  in  those  same 
proportions  in  their  general  exchange-value,  not  in  all  other 
things,  but  in  all  things  (or  in  all  l)ut  any  or  all  of  the  constant 

•"' Of  course  if  money  has  varied  in  general  exchange-value,  the  number  of 
things  that  are  constant  in  price  affects  the  condition  of  anything  otherwise  con- 
stant, hei'ausc  these  things  have  varied  in  general  exchange-value  with  money. 


THE    POSSIIULITY    OF    CONSTANCY  49 

things)  (Proposition  XXXIII.).  Thus,  while  the  variation  of 
anything  in  general  exchange-value  in  all  other  things  is  af- 
fected by  the  number  of  other  things  constant  in  general  ex- 
change-value, the  variation  of  anything  in  general  exchange- 
value  in  all  things  (or  in  at  least  all  the  variable  things,  including 
itself^  is  not  affected  by  the  number  of  other  things  remaining  con- 
stant in  general  exchange-value  (Proposition  XXXTV.).  The 
latter  part  of  this  proposition  is  apparent,  since  the  other  things 
that  are  constant  in  general  exchange-value  do  not  aifect  the 
general  exchange-value  of  the  one  thing  above  used,  by  com- 
parison with  which  the  variation  of  the  one  thing  may  be 
measured,  and  this  variation  may  be  measured  as  well  by  com- 
parison with  any  one  of  the  constant  things  as  by  comparison 
with  the  one  first  used.  The  distinction,  then,  is  needed  in 
order  to  carry  out  Proposition  XXIA^.,  which  shows  that  the 
relation  between  the  two  variations  in  general  exchange-value 
of  anything  is  affected  by  the  number  of  other  things;  for  then 
the  number  of  other  things  must  affect  the  two  variations  dif- 
ferently, and  if  the  number  of  other  things  that  are  constant  in 
general  exchange-value  does  not  affect  the  variation  of  any 
one  thing  in  general  exchange- value  in  all  things,  it  must  affect 
the  variation  in  general  exchange-value  in  all  other  things. 
More  particularly,  the  distinction  is  evident,  because  in  the 
calculation  by  which  the  constancy  of  the  constant  things  is 
determined  has  entered  the  variation  of  the  exchange-value  of 
the  thing  in  question  that  has  varied,  but  this  variation  is  ex- 
cluded from  the  calculation  by  which  its  variation  in  exchange- 
value  in  all  other  things  is  determined.  It  is  plain  that  only 
if  we  found  that  money,  for  instance,  remained  constant  in  ex- 
change-value in  all  other  things  but  one,  this  one  being  left  out 
of  the  calculation,  would  a  variation  of  this  one  relatively  to 
money  (indicated  by  its  price-variation)  measure  the  variation 
of  its  exchange-value  in  all  other  things.  And  what  would 
not  affect  the  exchange-value  in  all  other  things  of  a  thing  that 
varies  is,  not  something  remaining  constant  in  exchange-value 
in  all  things,  but  something  remaining  constant  in  exchange- 
value  in  all  but  that  one  thing,  that  one  thing  being  left  out  of 
4 


50  THE    CORRELATION    OF    EXCHANGE- VALUES 

the  calculation  of  the  constant  thing's  constancy  just  as  it  is 
left  out  of  the  calculation  of  its  own  variation  in  exchange- 
value  in  all  other  things.  In  exchange-value  in  all  things 
everything  is  measured  by  the  same  standard,  and  so  has  the 
same  variation  whether  it  be  directly  measured  in  relation  to 
all  things  or  whether  it  be  measured  in  relation  to  something 
whose  position  in  relation  to  all  things  has  already  been  deter- 
mined. But  in  exchange- value  in  all  other  things  the  standards 
are  different  in  every  measurement,  and  therefore  the  results 
do  not  fit  and  dove-tail  into  one  another  as  they  do  in  the  other 
case. 

The  last  Proposition  may  be  altered  into  this  :  If  in  measuring 
the  excJmnge-value  of  monet/  in  all  things  [including  itself)  we  find 
it  has  varied  to  a  certain  extent,  then  it  ought  to  be  indifferent  in  the 
calculation  whether  we  include  or  exclude  anything  ivhose  price  has 
varied  inversely  (Proposition  XXXV.).  In  other  words,  for 
instance,  if  in  measuring  money  Ave  find  that  the  general  level 
of  prices,  including  its  own,  which  never  varies,  has  risen  so  as 
to  be  /■  times  what  it  was  before,  wherefore  the  exchange- value 

in  all  things  of  money  has  fallen   by—,  then  anything  whose 

price  has  risen  so  as  to  be  r  times  what  it  was  before  ought  to 
be  indifferent  in  the  calculation  by  which  the  result  for  money 
was  obtained  ;  for  that  thing  has  remained  constant  in  exchange- 
value  in  all  things.  Furthermore  also,  If  in  measuring  the  ex- 
change-value of  money  in  all  other  things  (i.  e.,  in  all  commodities) 
we  find  it  has  varied  to  a  certain  extent,  then  it  ought  to  be  indif- 
ferent in  this  calculation  whether  ice  include  or  exclude  any  com- 
modity whose  price  has  varied  inversely  (Proposition  XXXVI.), 
since  this  commodity  retains  constant  exchange-value,  not  in 
all  things,  but  in  all  commodities,  and  so  is  constant  in  exchange- 
value  in  all  things  otiier  to  money,  and  the  addition  or  sub- 
traction of  such  a  commodity  to  or  from  the  standard  by  which 
the  variation  of  money  is  measured  does  not  alter  the  relation 
of  that  standard  to  money.'     Thus  the  standard  of  all  com- 

''  Here  if  r=  1,  indicating  constancy,  both  these  Propositions  fall  directly  under 
Proposition  XXXII. 


THE    POSSIBILITY    OF    CONSTANCY  51 

modities  (excluding  money)  may  take  the  place  of  the  standard 
of  all  things  (including  money). 

§5.  We  may  also  perceive  the  following  relations,  provided 
we  suppose  that  the  things  be  the  same  (or  similar  and  in  equal 
quantities)  at  all  the  periods  compared.  Then,  ichen  any  one 
thing  has  renuiined  constant  through  compensatory  variations  of 
others,  all  the  others  collectively  have  remained  constant  (Proposi- 
tion XXXVII.).  For  as  the  changes  in  the  other  things  have 
compensated  the  relations  of  this  one  thing  to  all  other  things, 
they  have  compensated  the  relations  of  all  the  other  things  to 
this  one  thing,  and  as  this  one  thing  is  constant,  they  have 
compensated  their  relations  to  something  constant,  and  there- 
fore themselves,  collectively,  are  constant.  Also,  according  to 
Proposition  XVIII.,  if  the  one  thing  had  risen  or  fallen  in 
exchange-value  in  every  other  thing  alike,  it  would  have  de- 
pressed or  raised  all  the  others  individually  somewhat  (and 
therefore  collectively  as  many  times  more) ;  therefore  if  it 
neither  rises  nor  falls  in  every  other  thing,  it  cannot  influence 
them  either  to  a  rise  or  to  a  fall,  and  they,  individually  (and 
collectively),  must  remain  constant.  Here,  however,  we  are 
supposing  the  one  thing  to  change  relatively  to  the  others  in- 
dividually, but  not  collectively  ;  yet  it  is  evident  that,  although 
the  others  change  relatively  to  it  individually,  they  cannot  do 
so  collectively.  It  follows  that  all  things  tahen  together,  namely 
all  things  permanent  over  all  the  periods  compared,  must  be  con- 
stant in  general  exchange-value  when  any  one  is  (Proposition 
XXXVIII.).  And  even  if  no  one  thing  has  remained  con- 
stant, yet  as  any  one  by  rising  depresses  all  the  others,  and 
each  of  the  others  less  in  proportion  to  their  numbers  (accord- 
ing to  Proposition  XIX.),  and  reversely  by  falling,  it  is  plain 
that  all  things  collectively,  provided  they  be  the  same  {or  similar 
and  in  equal  quantities)  at  all  the  periods  compared,  must  be  con- 
stant in  general  exchange-value  (Proposition  XXXIX.).  This 
principle  will  call  for  further  elucidation  later,  when  the  need 
of  the  proviso  will  be  proved,  and  it  will  be  shown  in  which 
kind  of  general  exchange-value,  when  there  are  variations,  the 
proposition  is  true. 


52  THE    CORRELATION    OF    EXCHANGE-VALUES 

§  0.  To  return  to  the  earlier  principles  about  the  possibility 
of  compensation,  it  follows  as  a  resume  of  them  that  it  is  pos- 
sible for  the  general  exchange-values  of  two  or  any  greater  number 
of  things  up  to  all  but  one  to  vary,  and  yet  for  the  general  ex- 
change-values of  all  the  others,  that  is,  of  any  number  of  tilings 
from  all  but  two  down  to  one,  to  remain  constant  (Proposition 
XL.).  Of  course,  for  this  result  to  happen,  the  variations  must 
be  in  certain  proportions,  which  it  is  our  object  to  discover.  As 
yet  we  only  know  the  general  possibility — and  not  even  that 
this  possibility  must  necessarily  always  exist,  but  that  it  may 
exist.  We  know  also  an  impossibility,  which  is,  that  if  any 
one  thing  alone  rises  or  falls,  it  is  impossible  for  anything  to 
retain  the  same  general  exchange- value,  since  the  counter- 
weight is  lacking. 

All  this  might  seem  to  apply  also  to  the  case  of  general  pur- 
chasing power.  It  can  be  so  if  this  term  be  identified  with 
exchange-value  in  all  other  things.  But  the  term  has  not 
always  been  used  in  this,  its  proper,  sense  ;  and  then  what  is 
here  proved  of  general  exchange-value  cannot  be  proved  of 
what  is  meant  by  "  general  purchasing  power."  This  subject 
will  be  discussed  later. 

The  statement  that  one  thing  is  remaining  constant  in  gen- 
eral exchange-value  while  other  things  are  varying  in  their 
general  exchange-values,  means  also,  as  we  have  seen,  that  this 
thing's  particular  exchange-values  in  those  varying  things  are 
varying  and  are  compensating  for  one  another.  This  statement 
may  be  true  of  money — of  which,  in  fact,  we  proved  it; — and 
as  the  particular  exchange-values  of  money  are  inversely  meas- 
ured by  prices,  it  follows  that  a  change  of  price,  or  many 
changes  of  prices  in  one  direction,  may  be  compensated  by  one 
or  many  changes  of  prices  in  the  opposite  direction,  with  the 
resultant  that  money  remains  constant  in  general  exchange- 
value,  the  changes  of  prices  neutralizing  one  another  and  yield- 
ing the  same  combined  influence  as  if  none  of  them  had  taken 
place.  Therefore,  although  we  have  seen  that  a  change  of  a 
single  price  cannot  directly  indicate  the  extent  of  the  thing's 
variation   in  exchange-value   in  all  things,  it  is  possible  for  a 


THE    POSSIBILITY    OF    CONSTANCY  53 

change  of  one  price  to  do  so  if  there  is  lit  least  one  other  cliange 
of  price  in  the  opposite  direction  to  the  proper  extent  for  coun- 
terbalancing ;  or  a  change  of  any  prices  may  do  so,  provided 
they  are  all  together  such  as  to  leave  the  exchange-value  of 
money  unchanged.  And  although  we  have  seen  that  a  change 
of  all  prices  alike  does  not  indicate  an  exactly  inverse  change 
of  money  in  exchange-value  in  all  things,  yet  such  a  change  of 
all  prices  but  one  may  do  so  if  that  one  changes  in  the  same 
direction  still  more  to  the  proper  extent  for  counterbalancing 
in  all  others  the  opposite  change  of  money  in  them.  We  are, 
however,  more  interested  in  the  former  case  of  money  being 
stable  in  general  exchange-value,  which  permits  particular 
changes  of  prices  directly  to  indicate  the  true  variation  in  ex- 
change-value in  all  things,  or  in  all  commodities,  of  the  things 
whose  prices  change.  This  can  be,  if  all  the  changes  of  prices 
counterbalance  one  another,  in  pairs  or  otherwise.  We  shall 
later  see  that  such  counterbalancing  is  not  wholly  left  to  acci- 
dent— which  if  it  were,  the  chances  might  be  as  small  of  its 
ever  happening  as  of  the  proverbial  letters  falling  out  of  a  bag 
and  composing  the  Iliad.  There  is  no  connection  between 
those  letters  :  there  is  close  connection  between  all  exchange- 
values. 


CHAPTER    III. 
ON  THE  MEASUREMENT  OF  GENEEAL  EXCHANGE -VALUE. 


§  1.  The  correlations  reviewed  in  the  preceding  chapter — the 
variation  of  other  things  in  exchange-valne  consequent  upon  a 
variation  of  any  one,  and  the  need  of  compensation  in  the  varia- 
tions of  other  things  in  order  to  keep  any  one  thing  constant — 
seem  to  point  to  a  peculiarity  in  exchange-value  such  as  to 
separate  it  off  from  other  quantities  that  are  subjects  of  mensu- 
ration. In  other  subjects  of  mensuration  we  consider  ourselves 
justified  in  thinking  we  have  some  things  possessing  fixed 
quantities  of  the  attribute  to  be  measured,  by  comparison  with 
which  we  can  measure  the  constancy  or  variation  of  the  things 
of  which  the  constancy  is  not  known  or  the  variation  is  suspected. 
Thus  in  measuring  lengths  at  different  periods  of  time  we  think 
we  can  take  the  circumference  of  the  earth  as  a  fixed  and  un- 
changing length,  or  even  as  such  the  length  of  any  carefully  pre- 
served stick  at  the  same  temperature,  or,  more  abstrusely,  the 
length  of  a  pendulum  swinging  in  a  second  at  a  given  spot.  But 
in  the  measurement  of  exchange-value  at  different  periods  we  find 
that  not  even  all  other  things  always  possess  a  fixed  exchange- 
value,  much  less  any  selected  one  from  amongst  them.  It  is 
as  if  we  lived  in  a  world  of  gases  and  were  seeking  to  measure 
length.  In  such  a  world,  in  which  a  change  in  size  of  one 
thing  would  involve  a  change  of  all  others,  perhaps  we  might 
succeed  in  the  attempt,  perhaps  not.  In  regard  to  exchange- 
value,  however,  the  case  is  not  hopeless.  In  fact,  the  way  out 
of  the  difficulty  has  already  been  indicated.  The  other  things, 
taken  collectively,  are  constant  in  certain  circumstances,  and  in 

54 


Cf)MPATlIS()X    WITH    OTHER    MEASUREMENTS  55 

others  tlieir  collective  variation  will  be  measurable  when  the 
variation  in  them  of  the  one  thino;  in  question  is  measured. 

The  most  striking  difference  is  this.  In  other  matters  we 
think  we  have  some  fixed  quantity  of  the  attribute  to  be  meas- 
ured, in  small  and  convenient  compass,  permanently  held  for 
us  by  some  material  thing,  which  fixed  quantity  of  the  attribute 
in  question  can  serve  as  a  small  unit  whereby  Ave  may  measure 
other  larger  or  smaller  quantities  of  the  same  attribute  in  other 
things.  In  matters  of  exchange  we  find  not  even  the  appear- 
ance of  any  fixed  quantity  of  exchange-value  being  permanently 
held  for  us  by  any  material  thing,  so  that  we  are  not  justified 
even  in  assuming  any  fixed  quantity  to  be  attached  to  any  ma- 
terial thing  we  know  of;  or,  in  other  words,  we  have  no  natur- 
ally provided  material  on  which  we  can  lay  out  a  small  unit  of 
exchange-value  serviceable  for  measuring  at  different  periods 
(or  places)  other  exchange-values  larger  or  smaller  than  it- 
Thus  other  measurable  things  at  least  seem  to  admit  of  our  form- 
ing absolutely  fixed  small  units,  by  comparison  with  which  we 
seem  to  be  able  to  speak  of  the  constancy  or  variation  of  abso- 
lute quantities  of  the  attril)ute  possessed  by  other  things.  And 
consequently  people  have  come  to  think  of  our  possessing  abso- 
lute measures  in  those  subjects.  But  in  exchanges  we  have  no 
even  apparently  fixed  small  unit,  so  that  we  seem  to  be  without 
power  of  measuring  the  variations  of  anything  in  absolute  ex- 
change-value. Exchange-values  are  recognized  as  only  rela- 
tive quantities ;  or  exchange-value  is  viewed  as  only  relatively 
quantitative.  And  so  exchange-value  has  been  distinguished 
as  merely  relative  from  other  quantities,  in  which  it  is  sup- 
posed we  have  something  absolute. 

§  2.  Hence  it  has  been  the  burden  of  the  song  of  many  econo- 
mists, who  sometimes  appear  to  find  pleasure  in  the  thought, 
that  the  measurement  of  general  exchange-value  through  the 
course  of  time  or  in  distant  places  is  impossible  because  nature 
has  not  provided  us  with  any  material  permanently  and  every- 
where possessing  the  same  general  exchange-value.  That 
nature  has  not  rendered  us  this  service  is  a  fact,  although 
we  have  seen  error  in   the  view  that   this  omission   is  due  to 


5G  MEASUREMENT    OF    GENERAE    EXOHAXGE-VALUE 

the  essential  impossibility  of  anything  having  fixed  exchange- 
value.  Others,  again,  have  despaired  of  measuring  general 
exchange-value  on  the  ground,  which  we  have  seen  to  be  in- 
sufficient as  well  as  incorrect,  that  it  is  not  an  attribute  in 
things  like  weight  or  color.  Still  others  have  denied  that  we 
can  measure  exchange-value  at  all,  for  the  reason  that  it  is  a 
relation,  and  that  we  cannot  measure  relations.^  None  of  these 
is  a  good  reason  for  despairing. 

The  only  real  difference  so  far  advanced  is  in  the  mistaken 
attitude  which  people  have  adopted.  In  other  subjects  of  men- 
suration we  have  nothing  more  absolute  than  what  we  have  in 
the  mensuration  of  exchange-value  itself.  The  differences 
whicli  exist  are  differences  of  degree,  and  not  of  kind. 

II. 

§  1.  That  exchange- value   is   a   relative  quantity,  is  not  a 

peculiarity  at  all.      All    quantities   are  relatively  quantitative. 

AVhen  a  branch  two  feet  long  grows  to  be  three  feet  long,  it  has 

changed  from  twice  to  thrice  the  length  of  a  foot-stick.     But 

^  F.  A.  Walker  :  "  Value  is  a  relation.  P^elations  may  be  expressed,  but  not 
measured.  You  cannot  measure  the  relation  of  a  mile  to  a  furlong ;  you  express 
it  as  8  to  1,"  3Ioney,  1877,  p.  288  (repeated  in  Bloney,  Trade  and  Industry,  1879, 
pp.  30,  60).  The  last  statement  is  merely  like  saying  that  we  cannot  measure, 
but  only  express,  the  relation  of  a  dollar  to  a  dime.  But  this  does  not  prevent  us 
from  measuring  other  exchange-values  by  dollars  or  dimes  just  as  we  measure 
other  distances  by  miles  or  furlongs. — Macleod :  "An  invariable  standard  of 
value  .  .  .  is  in  itself  absolutely  impossible  by  the  very  nature  of  things.  Because 
value  is  a  ratio :  and  a  single  quantity  cannot  be  the  measure  of  a  ratio.  A 
measure  of  lengtli  or  capacity  is  a  single  quantity  :  and  can  measure  other  single 
quantities.  .  .  .  But  value  is  a  ratio,  or  a  relation  :  and  it  is  utterly  impossible 
in  the  very  nature  of  things  tliat  a  single  quantity  can  measure  a  ratio,  or  a  rela- 
tion. It  is  impossible  to  say  that  a  :  b  :  :  x,"  Theory  'of  credit,  Vol.  I.,  p.  213. 
Length  is  no  more,  and  no  less,  a  single  quantity  tlian  exchange-value.  As  ex- 
change-value involvesa  relation  Ix'tween  things  in  exchange,  so  length  involves  a 
relation  between  points  in  space.  Neither  is  a  mere  relation,  the  one  being  the  dis- 
tance between  two  points,  the  other  a  power  in  one  thing  of  purchasing  another. 
Measurement  is  only  of  quantitative  relations.  In  measuring  length  we  say  that 
the  length  of  the  thing  in  question  is  to  afoot  (the  length  of  certain  sticks)  as.T  :  1. 
In  measuring  exchange-value  we  say  that  the  exchange-value  of  the  thing  in 
question  is  to  a  dollar  (the  exchange-value  of  certain  bits  of  metal)  as  x  :  1.  The 
processes  are  exactly  alike.  We  measure  one  relation  by  another  relation. — 
Laughlin  :  "  Since  value  is  a  relation  there  never  can  be  an  absolutely  perfect 
measure  of  it.  .  .  .  Length  is  positive,  and  not  a  relative  thing,"  op.  cit.,  p.  100. 
In  reality  it  is  only  because  value,  and  length  also,  are  relative  (quantitative), 
that  they  can  be  measured  at  all. 


THE    STANDARD    IX    8IMPTJ-:    MEXSUKATION  57 

the  foot-stick  itself  has  changed  from  a  half  to  a  third  of  the 
length  of  the  branch.  Only  here  we  say,  it  is  the  branch,  and 
not  the  foot-stick,  that  has  changed  in  length.  We  do  so, 
as  already  renaarked,  because  we  have  other  things  in  mind, 
and  the  foot-stick  is  the  one  which  keeps  its  length  unchanged 
relatively  to  them.  Of  course  it  cannot  be  supposed  that  we 
know  it  is  the  branch  which  grows  because  it  is  living,  and 
the  foot-stick  is  constant  because  it  is  dead,  wood.  For  we 
must  know  which  sticks  grow  and  which  are  constant,  before 
we  can  know  the  distinction  between  living  and  dead  wood. 
And  we  are  not  endowed  with  instinctive  or  innate  knowledge 
as  to  which  it  is  among  things  which  change  relatively  to  one 
another  that  is  constant  and  which  it  is  that  is  varying.  We 
acquire  this  knowledge  only  by  comparison  with  many  things, 
— and  comparison  is  the  primary  act  in  measuring.  Thus  even 
the  mensuration  of  length  is  a  relatively  quantitative  affair. 
We  measure  the  foot-stick  by  other  things  before  we  pronounce 
it  a  good  measure  for  measuring  other  things.^  There  is,  then, 
some  standard  in  this  mensuration  superior  to  the  material 
instrument  we  use  as  our  measure.  And  so  in  other  subjects 
of  mensuration.      What  is  the  nature  of  this  standard  ? 

§2.  In  measuring  some  attributes  or  qualities,  we  employ  a 
standard  foreign  to  them,  which  we  assume  to  be  a  fixed  stand- 
ard, so  far  as  they  are  concerned.  We  do  so  because  any  vari- 
ation that  might  occur  in  it  would  be  independent  of  variations 
in  them,  and  would  be  the  subject  of  another  measurement. 
Therefore  the  variations  in  them,  measured  in  this  standard, 
are  regarded  as  absolute  variations,  over  against  their  variations 
in  comparison  with  one  another,  as  relative  variations.  Yet 
this  view  of  the  absoluteness  of  the  variations  measured  by  this 
standard,  does  depend  upon  the  further  question  as  to  whether 
this  standard  can  be  found  to  be  really  fixed,  and  so  leads  us 
back  to  a  further  measurement.  Thus  in  the  general  subject 
of  value,  the  cost-value  of  a  thing  is  to  be  measured  by  the 
quantity  of  labor  required  to  produce  the  thing,  and  itsesteem- 

'  T.  ^laniuqiiiii :  "  We  do  not  measure  length  in  the  meter,  nor  capacity  in  the 
liter,"  Le  pr')h/ime  yno  net  a  ire,  lS7d,  i). '2S.  Of  course  while  usin;^  a  nieter-stick 
we  do  not  stop  to  measure  it.  But  we  confide  in  the  results  of  previous  measure- 
ments. 


58  MEASUREMENT    OF    GENERAL    EXCHAXGE-YALUE 

value  bv  the  esteem  in  Avhicli  it  is  held,  which  again  is  to  be 
measured  by  the  quantity  of  the  thing  forthcoming  relatively  to 
the  intensity  and  the  extension  of  human  desire  for  it.  There- 
fore, no  matter  how  much  a  thing's  exchange-value  in  all  other 
things  may  change,  its  cost-value  or  its  esteem-value  does  not 
change  unless  its  cost  of  production  or  its  rarity  changes,  and 
if  these  do  not,  it  may  be  looked  upon  as  absolutely  constant. 
But  now  the  question  is  renewed  as  to  Avhether  cost  of  produc- 
tion, or  quantity  of  labor,  can  be  absolutely  measured, — which 
involves  questions  about  both  absolute  time  and  absolute  feeling 
of  eifort ; — and  as  to  whether  quantities  of  matter,  or  masses, 
can  be  absolutely  measured.  In  all  cases  we  get  back  to  a 
measurement  of  an  attribute  or  quality  that  can  be  measured 
only  by  some  other  instance  of  the  same  attribute  or  quality, 
which  leads  us  to  the  question  how  this  is  to  be  measured,  and 
again  and  again  to  the  same  question,  wherefore  we  want  some- 
thing ultimate.  These  attributes  are  generally  the  first  ones 
we  start  with  the  attempt  to  measure,  such  as  length,  weight, 
and  a  few  others.  Among  them  is  general  exchange-value, 
although  this  has  been  late  in  calling  for  measurement.  They 
may  be  called  subjects  of  simple  mensuration. 

§  3.  In  subjects  of  simple  mensuration  we  have  nothing,  in 
the  sense  of  some  material  thing,  or  individual  instance  of  the 
attribute  measured,  that  is  ultimate  or  absolute — nothing  that 
we  can  know,  or  even  conceive,  to  be  constant  by  itself  alone, 
without  relation  to  something  else.  The  only  ultimate  standard 
we  have  here  is,  in  fact,  itself  a  relation.  It  is  the  relation  be- 
tween the  whole  of  the  quantities  to  be  measured  and  the 
quantities  themselves  as  its  parts.  And  the  only  constancy  or 
fixity  we  can  have  is  the  constancy  or  fixity  of  some  part  rel- 
atively to  this  whole.  We  see  this  most  clearly  in  that  most 
perfect  of  all  mensurations,  the  mensuration  of  angles.  Here 
the  unit  is  not  really  the  degree,  but  the  whole  circle,  of  which 
the  degree  is  only  a  fixed  part.  In  measuring  an  angle,  we 
compare  it  with  the  whole  circle  of  which  it  is  a  part.  And 
an  angle  is  fixed,  or  constant,  as  it  keeps  the  same  relation  to 
this  whole.     And  so  in  measuring  the  length  of  anything,  we 


THE    STANDARD    IX    SIMPLE    MENSURATION  59 

are  really  comparing  it  with  a  larger  whole — if  not  the  whole 
extended  universe,  the  whole  earth,  or  at  least  the  whole  of 
things  we  see  around  us.  Thus  when  a  branch  and  a  foot- 
stick  change  in  length  relatively  to  each  other,  we  perceive  that 
the  branch  changes  in  length  relatively  to  the  earth,  and  we 
do  not  perceive  any  change  in  the  foot-stick  relatively  to  the 
earth.  To  be  siwe,  the  foot-stick  has  changed  relatively  to  one 
other  thing.  But  we  know  of  other  things  becoming  smaller 
at  the  same  time,  in  comparison  with  which  the  foot-stick  has 
become  larger,  and,  among  the  infinitude  of  things  with  infin- 
itely different  lengths,  we  are  unable  to  perceive  any  change  in 
the  length  of  the  foot-stick  (at  the  same  temperature)  compared 
with  the  whole  of  things.  If  we  did  (as  we  do,  in  fact,  at 
different  temperatures),  we  should  have  to  say  that  the  foot- 
stick  had  changed  in  length. 

§  4.  The  fixity  is  not  necessarily  in  the  Avhole  itself  inde- 
pendently of  its  parts,  nor  in  the  parts  independently  of  the 
whole.  The  only  fixity  we  can  have  knowledge  of  is  in  the 
relation  between  the  parts  and  the  whole. 

It  is  not  necessarily  in  the  whole  itself;  for  our  universe 
might  expand  or  contract  in  size  without  our  knowing  it,  so 
long  as  all  particular  things,  including  our  bodies,  kept  the 
same  relationship  to  the  whole,  as  we  perceive  it, — and  who 
knows,  or  who  cares,  whether  the  universe  of  material  things  is 
one  day  half  as  large  as  another  in  so-called  absolute  size  ? 
Such  a  change  would,  after  all,  have  to  be  a  change  in  relation 
to  something  else  than  our  material  universe — say,  another 
material  universe,  or  something  called  absolute  space.  Then 
if  such  a  change  actually  did  take  place,  what  we  should  desire 
for  our  measure  of  length  would  be,  not  absolute  permanence — 
permanence  in  relation  with  that  extra-mundane  thing, — but 
relative  permanence,  in  relation  to  our  own  universe  or  world. 
What  happens  to  our  universe  and  its  parts  in  comparison  with 
something  else  beside  it,  in  no  wise  concerns  us,  and  we  do  not 
care  to  measure  this  relation.  Moreover,  ev'en  if  we  did  find 
such  a  change,  we  could  not  think  of  it  as  being  more  in  our 
universe  tiian  in  that  other  thing,  except  by  comparison  with 
still  another  thing ;  and  so  on  without  end. 


60  MEASUREMENT    OF    GExVERAL    EXCHANGE- VALUE 

Nor  is  the  fixity  necessarily  in  tlie  parts.  In  one  physical 
theory  our  material  universe  is  conceived  to  be  composed  of  ex- 
tended atoms,  which  do  not  touch  one  another.  Now  suppose 
that  all  atoms  are  approaching  toward  or  receding  from  one 
another,  that  is,  that  the  distances  between  the  atoms  are  en- 
larging or  decreasing  compared  with  the  distances  through  the 
atoms.  Such  a  change,  say  of  contraction,  may  be  stated  in 
two  (among  five)  typical  ways  :  the  one,  that  the  atoms  are  per- 
manent in  size,  and  the  intervening  spaces  are  decreasing ;  the 
other,  that  the  intervening  spaces  (between  the  centers)  are  per- 
manent, and  the  atoms  are  enlarging — in  each  case  relatively 
to  something  else  outside  our  universe.  By  the  physicist  the 
former  of  these  interpretations  is  the  more  likely  to  be  adopted, 
because  it  fits  in  better  with  his  preconceptions,  which  generally 
do  not  take  into  account  what  is  meant  by  permanence  in  size 
— namely,  permanence  in  relation  to  some  other  size.  Yet,  if 
we  could  actually  see  this  change  going  on,  -vve  should  be  as 
likely  to  say  that  the  atoms  are  growing  larger  as  to  say  that 
the  universe  is  growing  smaller.  Practically,  however,  so  long 
as  we  remain  out  of  touch  with  these  smallest  things,  if  the 
things  which  are  the  smallest  for  us,  and  all  the  things  which 
we  see  and  feel,  including  our  bodies,  grew  smaller  compared 
with  the  atoms  in  them  in  exactly  the  same  proportion  (or  on 
the  average,  allowing  for  relative  changes  in  some  things,  such 
as  go  on  anyhow),  so  that  they  all  (with  these  exceptions)  re- 
tained the  same  relations  to  the  whole,  we  should  want  our 
measure  of  length,  at  least  for  all  practical  purposes,  to  decrease 
with  the  rest,  so  as  to  retain  the  same  relation  to  the  whole 
as  it  had  before,  the  sameness  of  this  relation  being  our  stand- 
ard, although  then  our  measure  of  length  would  decrease  in 
size  relatively  to  the  atoms,  w^hich  are,  according  to  the  hy- 
pothesis, outside  our  visible  and  tangible  world.  Then,  prac- 
tically for  us,  measured  by  the  standard  which  we  use,  it  would 
be  the  atoms  that  are  growing  in  size.' 

§  5.   In  a  similar  manner,  when  one  thing,  say  a  branch, 

^  Probably  the  measurement  of  length  by  the  pendulum  would  indicate  a 
change.  But  then  we  would  be  at  a  loss  as  to  whether  the  change  were  in  our 
usual  mensuration  of  length,  or  in  our  mensuration  of  time. 


THE    STANDANT)    IX    8IMPI.?:    Mf^NSUIlATIOX  61 

changes  in  length  before  our  eyes  in  comparison  with  other 
things,  it  would  be  possible  to  assume  that  the  branch  has  re- 
tained tlie  same  size  and  all  the  rest  of  our  universe  has  grown 
smaller, — that  is,  relatively  to  something  else  beside  the  uni- 
verse. But  if  we  adopted  this  view, — and  if  it  were  true, — 
we  should  still  want  our  measure  of  length  to  go  with  the  rest 
of  our  universe,  instead  of  remaining  constant  with  this  one 
thing  in  its  mystical  connection  with  something  else  about  which 
we  know  nothing  beyond  its  agreement  with  this  thing.  We, 
therefore,  adopt  the  opposite  course,  and  think,  not  really  of 
our  universe  as  unchanging,  but  of  the  unchanging  measure  of 
length  as  the  one  which  remains  permanent  in  relation  to  our 
universe,  and  as  the  branch  is  changing  in  relation  to  our  uni- 
verse and  to  tliis  measure,  we  think  of  the  branch  as  changing 
in  length.  It  is  not  that  we  bring  in  any  theory  of  probabilities, 
and  argue  that  it  is  more  probable  that  the  branch  changes  and 
that  the  universe  does  not.  What  we  see  is  that  the  branch 
does  change  relatively  to  the  whole,  and  to  other  things  which 
are  not  seen  to  change  relatively  to  the  whole.  While  it  is 
doing  so,  the  universe  and  many  things  in  it  do  change  equally 
much  relatively  to  it,  but  we  do  not  see  that  the  universe  does 
change  at  all  relatively  to  the  things,  the  foot-sticks,  Avhich  we 
have  chosen  for  our  measures.  Hence,  we  assign  the  change 
to  the  branch,  and  not  to  the  foot-stick  or  to  the  universe. 
What  we  measure  is  a  fact,  not  a  probability. 

What  is  here  said  of  length,  is  true  also  of  other  subjects  of 
simple  mensuration.  So,  for  instance,  in  the  case  of  mass  of 
matter.  A  body  has  permanent  mass,  if  it  always  has  the 
same  quantity  in  relation  to  the  whole  quantity  in  our  universe. 
This  is  our  ultimate  conception  of  permanence  of  mass ;  for  we 
know  nothing  about  any  other  permanence  of  the  mass  of  the 
whole  universe  itself,  or  of  any  of  its  parts,  which  perma- 
nence, if  it  existed,  would  have  to  be  in  relation  to  something 
else.  If  new  matter  were  injected  into  all  things  in  the  same 
proportion,  we  might  not  perceive  any  change,  and  if  some- 
how we  did  learn  of  it,  we  should  be  glad  that  our  measures  of 
mass  were  increased  along  with  the  rest.  Similarly  in  the  case 
of  force,  in  its  simplest  forms. 


62  MEAST'KEMENT    OF    GENERAL    EXCHANGE-VALUE 

But  if  any  of  these  liypothetical  and  miraeulous  changes 
took  place  in  a  part,  say  a  half,  of  our  world,  especially  if 
scattered  about,  we  should  be  at  a  loss  where  to  place  it 
(whether  to  conceive  of  it  as  an  increase  in  the  one  half,  or  as 
a  decrease  in  the  other),  and  should  distribute  it  over  the  whole  ; 
and  then  our  measure  would  be  some  size,  or  some  weight, 
which  has  the  same  relation  to  the  whole  as  our  old  measures 
had. 

§  6.  In  all  these  subjects  we  are  somewhat  careless  of  the 
widest  standard  theoretically  possible,  and  content  ourselves 
with  the  most  practicable  and  the  most  prominent.  Thus  our 
ordinary  measures  are  measured,  not  by  their  relation  to  the 
whole  material  universe,  but  by  their  relation  to  the  earth,  or 
even  to  particular  regions  on  the  earth,  and  have  even  been 
measured  with  more  especial  reference  to  the  sizes  of  our  bodies. 
If,  therefore,  such  changes  as  just  supposed  were  to  take  place 
in  the  outer  stellar  regions,  we  should  probably  pay  no  atten- 
tion to  them  as  regards  our  measures  of  length  or  of  mass  ; 
for  the  relation  between  our  measures  and  our  standard  for 
them,  the  earth,  would  be  unchanged.^  Again,  if  something 
were  miraculously  increased  on  our  earth,  we  should  probably 
think  it  had  received  matter  from  the  outside,  and  not  alter 
our  conceptions.  But  if  we  all  woke  up  some  fine  morning 
and  found  half  the  things  with  which  we  are  familiar  become 
larger  compared  with  the  other  half,  and  this  other  half  there- 
fore smaller  compared  with  those,  then,  even  if  all  our  meas- 
uring sticks  happened  to  remain  alike  and  were  constant  with 
the  one  or  the  other  of  these  sets,  we  should  have  to  adapt  our 
measure  of  length  to  some  average  of  all  these  changes ;  for 
to  say  that  our  measuring  sticks  had  remained  constant,  would 
be  to  'take  the  half  of  things  witli  which  they  continue  to 
agree  as  the  constant  ones,  although  we  should  have  no  more 

'  At  all  events  in  our  measurements  of  things  on  the  earth.  For  in  astronomy 
we  have  to  make  use  of  other  l)odies  beside  the  earth  in  our  measurement  of  dis- 
tance. It  is  coneeivable  that  the  astronomical  and  the  geographical  miles  could 
vary.  If  the  world  were  still  molten  and  were  contracting,  we  being  salamanders 
that  live  in  fire,  we  might  find  this  to  be  the  case.  Perhaps  we  could  find  it  even 
now  if  our  measurements  were  sufficiently  accurate. 


THE    STANDARD    IN    SIMPLE    MENSURATION  63 

right  to  do  tliis  than  to  take  the  other  half  as  the  constant 
ones.  What  we  should  desire  for  our  standard  of  length  would 
be  some  portion  of  the  lengthened  things,  and  some  addition 
to  the  shortened  things,  that  has  the  same  relation  to  the  whole 
as  our  standard  measures  had  before. 

§  7.  To  be  sure,  such  variations  are  not  found  to  take  place 
in  the  lengths  and  in  the  weights  of  things  on  the  face  of  the 
earth, — which  is  why  we  should  regard  them  as  miraculous  if 
such  changes  should  suddenly  begin  to  take  place.  When  we 
open  our  eyes  every  day,  what  we  see  contains  many  things — 
mountains,  plains,  rocks,  buildings  and  land-marks  of  thousands 
of  kinds  and  descriptions, — that  do  not  appear  to  alter,  do  ap- 
pear to  remain  the  same,  in  their  relative  positions  and  in  their 
relative  distances  or  lengths.  And  many  things,  when  they  are 
left  to  themselves,  appear,  when  we  weigh  them,  to  have  the 
same  weight  (relatively  to  many  other  similar  things)  always. 
Upon  these  we  hit  for  our  standard.  Thus  the  foot-stick,  or 
the  pound  weight,  which  we  use  for  measuring,  is  one  which 
we  find  not  to  v'ary  in  relation  to  other  things — to  be  one 
among  the  things  that  do  not  apparently  vary  relatively  to  one 
another.  Yet,  although  this  constancy  of  so  many  things  rela- 
tively to  one  another  obscures,  it  does  not  alter,  the  principle 
of  mensuration.  It  is  only  because  the  foot-stick  (at  the  same 
temperature)  is  not  perceived  to  vary  relatively  to  the  whole  of 
things,  that  we  are  justified  in  thinking  of  it  as  a  constant 
measure.  We  are  not  justified  in  thinking  so  simply  because 
there  are  some  things  which  keep  constant  relations  of  length 
amongst  themselves.  There  must  be  enough  of  these  to  form 
a  whole,  in  which  not  only  our  measures,  but  all  the  things  in 
which  we  are  interested  are  included.  And  in  this  whole  must 
be  included  also  the  things  which  do  vary.  And  if  these  things 
which  do  vary  are  not  to  affect  the  measure  by  its  variations 
relatively  to  them,  it  must  be  because  of  compensations  in  their 
variations.  For  instance,  if  all  the  things  that  vary  in  length 
varied  only  by  growing  larger  relatively  to  the  things  that  are 
constant  amongst  themselves,  which  then  are  always  becoming 
smaller  relatively  to  those  other  things,  then  (supposing  those 


64    MEASUREMENT  OF  GENERAL  EXCHANGE-VALUE 

other  tilings  to  be  sufficiently  important  to  interest  us)  we  should 
not  consider  the  things  stationary  amongst  themselves  to  be  a 
good  standard  by  which  to  measure  the  others.  We  should 
want  to  include  tiie  others  in  the  whole,  and  the  measure  would 
have  to  increase  partly  with  the  increase  of  the  others.  We  are 
saved  from  this  need  only  by  the  fact  that  the  things  which  do 
vary  in  length  relatively  to  the  things  stationary  amongst  them- 
selves, become  both  larger  and  smaller  in  about  the  same  pro- 
portions. Trees  which  grow,  also  decay  and  fall  or  are  chopped 
to  bits,  and  disappear,  as  others  appear.  Animals  increase,  and 
die  away.  Clouds  form,  and  fade.  Heat  expands,  and  cold 
contracts.  It  is  only  because  our  foot-sticks,  which  remain 
steady  in  relation  to  the  things  that  do  not  change  amongst 
themselves,  remain  steady  also  in  relation  to  all  these  increasing 
and  decreasing  things,  on  the  whole,  to  all  appearance,  that 
they  are  suitable  measures  of  length. 

§  8.  Now  the  exchange- value  of  everything  forms  part,  with 
the  exchange-value  of  everything  else,  of  a  whole.  By  com- 
parison with  this  whole  we  ought  to  be  able  to  reach  the  same 
kind  of  fixity  in  this  subject  as  in  other  quantitative  matters. 
To  be  sure,  we  are  here  left  without  the  help  of  things  that  re- 
main constant  relatively  to  one  another.  In  the  economic 
world  there  are  no  mountains,  plains,  rocks,  buildings  and  other 
land-marks  that  retain  the  same  exchange-values  relatively  to 
one  another.  We  have  only  things  that  change  relatively  to 
one  another.  Yet  this  difference  is  only  a  difference  of  degree, 
and  not  of  kind.  The  relative  fixity  in  length  of  land-marks 
inter  se  is  a  help,  and  a  great  help,  but  not  a  requirement,  in 
the  mensuration  of  length.  The  standard  is  the  relation  of  a 
part  to  the  whole — or  to  a  practicable  whole.  This  standard 
we  can  have  in  the  measurement  of  exchange-value,  as  well  as 
of  other  attributes.  The  principle  of  simple  mensuration  is  the 
same  in  all  cases.  In  measuring  exchange- value  what  we  need 
is  to  form  a  proper  conception  of  the  whole  of  exchange-values. 
This  done,  we  shall  be  able  to  compare  with  it  the  exchange- 
value  of  anything,  in  order  to  find  whether  this  exchange-value 
is  constant  or  varving  in  relation  to  the  whole,  and  how  much 


THE    STANDARD    IX    SIMPLE    ME.\Sl'I{ATI(X\  05 

varying.  If  it  be  constant,  not  only  the  whole  including  it, 
but  the  whole  of  all  the  rest  of  things,  will  be  constant  rela- 
tively to  it.  If  it  rise,  the  latter  whole  will  fall  ;  and  reversely  ; 
diverging  from  the  fall  or  rise  of  the  whole  including  the  thing 
in  a  definite  proportion.  It  will,  then,  be  sufficient  to  be  able 
to  compare  the  exchange-value  of  any  one  thing  with  the  ex- 
change-value of  all  the  others. 

If  we  succeed  in  this,  the  absence  of  any  particular  thing 
that  naturally  keeps  its  relation  to  the  whole  permanently  un- 
changed will  in  no  wise  prevent  us  from  reaching  satisfactory 
results.  We  shall  have  a  method  of  measuring  exchange-value, 
in  the  place  of  a  measuring  instrument.  Instead  of  having  a 
bit  of  material  which  keeps  the  unit  tolerably  constant,  we 
shall  be  able  to  lay  out  a  unit  of  exchange-value  upon  materials 
that  vary  in  exchange-value.  In  doing  so,  however,  we  shall 
do  only  what  is  done  in  all  other  matters  of  mensuration  when 
the  greatest  possible  accuracy  is  desired.  The  metal  used  as 
bearer  of  the  unit  of  length  does  not  remain  constant  under 
changes  of  temperature,  and  the  engineer  must  allow  for  its 
variation,  often  using  as  a  foot  a  length  which  is  not  the  length 
of  his  foot-stick.  But  more  than  this.  The  possibility  exists 
that,  although  nature  does  not  provide  us  with  a  material  un- 
varying in  exchange-value,  we  shall  be  able  to  make  such  a 
thing  for  ourselves,  with  tolerable  exactness,  and  so  be  in  pos- 
session of  a  dollar-bit  of  metal  or  paper  comparable  with  a  foot- 
stick  of  wood  or  metal. 

That  exchange- value  is  not  something  absolute,  is,  therefore, 
no  objection  against  our  being  able  to  measure  it  with  as  great 
precision  as  we  can  attain  to  in  other  subjects  of  mensuration, 
since  in  no  subject  of  mensuration  is  the  attribute  itself,  which 
we  measure,  anything  absolute.*     And  what  we  have  of  fixity, 

*  J.  Gamier:  "  Unfortunately,  all  value  being  essentially  variable,  it  follows 
that  there  cannot  be  an  invariable  unit  of  value,  and  that  we  cannot  estimate  the 
absolute  magnitude  of  the  value  of  things,  but  only  their  relative  and  comparative 
magnitude.  .  .  .  The  value  of  this  sum  of  money  [by  which  we  attempt  to  meas- 
ure the  value  of  a  house]  is  not  value  existing  by  itself,  abstraction  being  made  of 
all  comparison,  and  we  can  form  an  idea  of  it  only  by  comparing  it  with  all  the 
things  we  can  obtain  in  exchange  for  it  [including  the  house  itself] .  ...  It  fol- 
lows therefore  from  the  internal  nature  of  value  that  the  search  after  a  (mathe- 

5 


66  MEASUREMENT    OF    GENERAL    EXCHAN(4E-YALUE 

or  of  absoluteness  in  this  sense,  in  otlier  measurements,  we  have 
also  in  the  case  of  exchange-value,  so  far  as  its  relativity  is  con- 
cerned— namely,  fixity  of  the  relation  between  a  whole  and  its 
parts. ^ 

§  9.  It  is  strange  how  great  is  the  fondness  for  absoluteness, 
and  how  great  the  dislike  of  relativity.  In  subjects  wholly 
relative,  where  much  was  once  carelessly  thought  to  be  absolute, 
people  are  fain  to  retain  the  absolute  in  some  nook  or  corner, 
letting  it  in  by  the  back-door  after  driving  it  out  in  front. 
The  eminent  mathematician  and  economist,  Cournot,  drew  the 
distinction  that  we  cannot  have  an  absolute  exchange-value, 
but  we  can  have  an  absolute  variation  of  exchange-value.® 

Such  a  distinction  is  due  to  an  ambiguity  in  the  term  "  abso- 
lute." The  literal  meaning  of  "  absolute  "  is  "  without  relation 
to  anything  else."  It  has  acquired  the  meaning  of  "  really  fixed 
or  permanent,"  or  "  really  invariable  in  its  relations."  Besides 
which,  the  adverbial  form  is  often  used  in  a  merely  intensive 
sense,  as  when  we  say  "  absolutely  all,"  meaning  "  all  without 
any  exception  whatsoever."  Plainly  it  is  only  the  second,  the 
acquired,  sense  that  has  any  importance  in  matters  of  mensura- 
tion. Nothing — not  even  the  universe,  or  *' absolutely  "  all 
things — can  be  absolutely  variable  or  absolutely  constant  in 
size  or  in  any  other  quantitative  attribute,  in  the  original  sense. 

niiitically)  exact  standard  or  meter  of  value  is  impossihle,"  op.  cit.,  p.  290  (the 
italics  in  the  original).  This  passage  finds  fault  with  the  mensuration  of  value 
for  ))eing  able  to  be  no  more  than  it  ouglittobe,  a  measurement  of  the  relative  or 
comparative  magnitude  of  values.  It  also  makes  the  mistake  of  implying  that 
the  meter,  the  unit  of  length,  is  not  only  a  mathematically  exact,  but  also  an  ab- 
solute, standard,  existing  by  itself,  with  abstraction  of  all  comparison.  Thus  the 
fault  found  with  the  mensuration  of  exchange-value  is  that  it  is  not  what  the 
mensuration  of  length  also  is  not. 

^  "We  have  seen  that  esteem-value  is  measured  by  esteem,  and  cost-value  by 
labor-cost.  We  may  notice,  in  passing,  that  esteem  and  cost  are  to  be  measured 
in  the  same  way  as  above  described.  We  have  a  certain  esteem  for  all  the  things 
we  possess.  If  the  things  we  possess  become  more  numerous,  the  size  of  each  com- 
pared with  the  whole  decreases.  So  our  esteem  for  each  decreases.  And  with  it 
each  thing's  esteem-value.  Reversely  if  our  possessions  grow  less.  Then  every 
single  one  becomes  larger  in  relation  to  the  whole;  consequently  it  grows  in 
esteem,  and  in  esteem-value.  Similarly  with  cost-value.  If  an  hour's  work 
comes  to  produce  more  things,  each  of  these  becomes  smaller  in  relation  to  the 
whole  product  :  it  falls  in  labor-cost,  and  in  cost-value. 

'■  Op.  oil.,  p.  22. 


TWO    GEXKRAI.    STANDARDS  67 

But,  ill  the  acquired  sense,  a  thing  can  be  absolutely  permanent 
as  well  as  absolutely  variable  in  relation  to  the  standard 
adopted — and  if  a  thing  be  found  absolutely  permanent,  say  in 
size,  its  size  might  just  as  well  be  spoken  of  as  absolute — in  this 
secondary  sense.  Exactly  so  with  exchange-value.  Cournot 
himself  distinguished  ''absolute  variation  "  from  "  relative  vari- 
ation "  only  because  when  two  things  change  in  exchange-value 
relatively  to  each  other,  we  can  conceive  of  the  change  as  being 
wholly  in  the  one."  He  apparently  failed  to  see  that  in  this 
case  we  are  merely  comparing  each  of  the  things  with  all  things, 
and  the  change  is  wholly  in  the  one  only  because  this  one  is 
changing  relatively  to  all  other  things,  while  the  other  is  not 
changing  relatively  to  all  other  things.  The  variation  of  the 
one,  then,  is  as  relative  as  the  permanence  of  the  other.  And 
if  the  variation  of  the  one  can  be  called  absolute,  because  of  the 
comparison  with  the  true  standard,  so  can  the  permanence  of 
the  other.  And  if  we  can  say  a  thing  can  be  absolutely  per- 
manent in  exchange-value,  we  can  equally  well  say  it  is  perma- 
nent in  absolute  exchange-value.  All  this  use  of  the  term 
"  absolute,"  however,  should  be  avoided  when  dealing  with 
quantities,  quantities  being  wholly  relative.  For  to  use  it  in 
the  acquired  sense  involves  the  risk  of  importing  into  it  also 
the  literal  sense  of  "  without  relation." 

III. 

§  1.  Even  the  distinction  between  two  kinds  of  general  ex- 
change-value— its  division  into  exchange- value  in  all  other 
things  and  exchange-value  in  all  things — is  not  peculiar  to  our 
subject.  Thus  in  our  extended  universe  if  something  in  chang- 
ing in  its  size  does  not  aifect  the  size  of  the  whole,  this  distinc- 
tion exists ;  for  if,  for  example,  the  thing  were  reduced  to  half 
its  size  in  relation  to  the  whole,  it  would  be  slightly  more  than 
halved  in  relation  to  the  other  things.  Or  again,  if  in  chang- 
ing in  its  size  it  affects  the  size  of  the  whole  by  so  much,  then, 
if  for  example  it  were  halved  compared  with  what  the  whole 
used  to  be,  it  would  be  halved  compared  with  all  other  things, 

l  Op.  cit.,  p.  18. 


68     MEASUREMENT  OF  GENERAL  EXCHANGE-VALUE 

but  not  quite  halved  compared  with  all  things.     The  distinction 
may  be  more  plainly  seen  in  the  case  of  motion,  which  yields  a 
closer  analogy  with  our  subject — already  used  in  a  partial  way. 
In  a  finitely  extended  universe  of  points,  the  whole  of  which 
we  cannot  conceive  to  move,  since  there  is  nothing  in  relation 
to  which  it  could   move,  if  all  others  of  its  points  retain  their 
positions   unchanged    relatively  to  one  another,  the   apparent 
motion  of  a  single  point  a  certain  distance  (compared,  say,  with 
the  diameter  of  the  whole)  will  be  its  real  motion.     This  is  its 
motion  relatively  to  all  other  points.      Again,  in  a  finitely  ex- 
tended material  universe,  in  which  its  parts  have  attractions  so 
that  for  the  whole  there  is  a  center  of  gravity,  which  we  can- 
not conceive  to  move  (except  in  relation  to  other  parts  of  the 
universe),  since  its  motion  would  represent  the  combined  mo- 
tion of  the  whole,  and  we  cannot  conceive  of  the  whole  as  mov- 
ing, for  the  same  reason  as  before;  then,  all  other  bodies  retaining 
their  positions  unchanged  relatively  to  one  another,  the  motion  of 
a  single  body  a  certain  distance  through  the  others  in  a  certain  di- 
rection would  displace  the  center  of  gravity  in  the  whole  a  slight 
distance  in  the  same  direction,  and  therefore,  this  center  being 
conceived  as  unmoved,  it  would  cause  the  rest  of  the  universe  to 
move  a  slight  distance  in  the  opposite  direction  (relatively  to  its 
center  of  gravity),  and  its  own  apparent  motion  (relatively  to  all 
the  others)  will  not  be  its  real  motion  (in  relation  to  the  center 
of  gravity  of  the  whole),  which  will  be  slightly  less  in  the  same 
direction.     This  is  obviously  its  motion  relatively  to  all  things, 
including  itself.     Thus  in  the  former  case,  or  in  the  latter  also 
if  we  conceive  of  the  motion  only  as  compared  with  all  the 
other  things,  that  motion  is  comparable  with  the  rise  or  fall  of 
the  exchange-value  of  a  single  exchangeable  thing  in  all  oilier 
things ;  and  in  the  latter  case,  what  we  regard  as  the  body's 
real  motion  is  comparable  with  the  rise  or  fall  of  the  exchange- 
value  of  a  single  exchangeable  thing  in  all  things.     In  both  the 
subjects,   when  our  attention    is  called  to  the  distinction,  we 
regard  the  latter  point  of  view  as  the  truer  one — the  one  alone 
suitable  for  "all  measurements, — but  the  former  as  also  a  pos- 
sible one.      In  the  case  of  motion  a  body  cannot  move  without 


TWO    GEXEKAL    STANDARDS  G9 

pushing  or  pulling  something  else  in  the  opposite  direction,  and 
if  that  something  else  retains  the  same  position  relatively  to  all 
other  things,  all  these  other  things  must  have  been  shoved  in 
the  opposite  direction,  though  only  to  a  very  slight  extent,  so 
that  the  motion  of  the  body  in  question  is  not  so  far  in  reality 
as  it  is  in  relation  to  them,  since  they  are  moving  slightly  in 
the  opposite  direction.  And  so  with  the  movement  of  an  ex- 
changeable thing  upward  or  downward  in  exchange- value  rela- 
tively to  all  other  things  :  it  pushes  them  somewhat  in  the 
opposite  direction,  Avherefore  its  motion  relatively  to  them  is 
greater  than  it  really  is,  as  they  are  moving  in  the  opposite  di- 
rection.^ For  convenience,  however,  especially  as  the  opposite 
motion  of  all  the  other  things  is  infinitesimally  small  (though 
less  so  in  economics  than  in  physics),  we  can,  if  we  choose,  re- 
gard all  the  other  things,  or  even  only  those  of  them  which  re- 
main unchanged  relatively  to  one  another,  as  our  standard,  and 
measure  not  merely,  as  we  do,  the  motion  of  a  body  by  its  re- 
lation only  to  all  or  even  to  some  other  things,  but  also  the 
variation  of  a  thing's  exchange-value  merely  by  its  relation  to 
the  other  things. 

§  2.  When  ^ve  have  chosen  which  method  we  shall  adopt, 
and  what  shall  be  our  standard,  there  is  of  course  no  occasion 
for  employing  in  our  measurements  the  law  of  probabilities — 
as  was  asserted  also  in  this  connection  by  Cournot.^  We  do 
not  say  :  it  is  more  probable  that  all  the  other  things  have  re- 
mained stationary  than  that  this  one  has  stood  still  and  they 
moved ;  or,  it  is  more  probable  that  all   things  have   together 

1  If  the  moving  body  pushed  something  else  equally  heavy  equally  far  in  the 
opposite  direction,  or  several  things  appropriately  lesser  distances,  the  motion  of 
this  body  (no  longer  the  only  one  moving  compared  with  the  rest)  would  be 
compensated,  so  that  its  apparent  motion  compared  with  all  the  common  other 
things  which  have  remained  unchanged  amongst  themselves  would  be  its  r«il 
motion  compared  with  all  things  (including  itself),  while  those  common  other 
things  would  remain  unchanged  also  in  their  real  positions.  The  similarity  with 
the  case  of  exchange-value  as  exposited  in  Proposition  XXXIII.  is  manifest. 
There  is  only  one  difierence,  which  gives  greater  simplicity  to  the  case  of  exchange- 
value.  The  motion  of  bodies  in  space  may  be  in  three  dimensions ;  the  motion  of 
things  in  exchange-value  can  be  only  in  one  dimension — hence  always  only  in 
either  the  same  or  opposite  directions.  In  exchange-value  there  is  no  parallelo- 
gram of  forces,  except  as  this  is  reduced  to  a  straight  line. 

^  Op.  cit.,  pp.  15-16.     Similarly  Bourguin,  B.  132,  p.  24. 


70  MEASUREMENT    OF    GENERAL    EXCHANGE-VALUE 

remained  stationary,  wherefore  both  this  and  the  others  have 
moved  relatively  to  the  whole.  But  having  adopted  our  point 
of  view,  we  simply  measure,  as  best  we  can,  what  we  see  hap- 
pening before  us.  And  our  point  of  view  itself  in  these  matters 
we  adopt,  not  by  any  use  of  the  law  of  probabilities,  but  because 
the  myriad  interrelations  which  do  not  change,  or  which  do  not 
change  on  the  average,  make  more  impression  on  us  than  the 
particular  ones  which  do  change. 

IV. 

§  1.  Having  found  these  points  of  resemblance  between  the 
mensuration  of  exchange-value  and  the  mensuration  of  other 
ultimate  quantitative  attributes,  let  us  turn  to  a  difference  that 
is  of  considerable  moment. 

The  ultimate  standard,  consisting  of  a  relation  between  the 
parts  and  the  whole  composed  of  them,  would  seem  to  demand, 
for  its  perfection,  that  the  whole  should  be  the  same,  or  exactly 
similar,  whole  at  both  the  periods,  or  at  both  the  places,  between 
which  the  comparison  is  instituted.  This  requirement  is  ob- 
served in  the  mensuration  which  we  have  already  noticed  as 
being  the  most  perfect,  namely  the  mensuration  of  angles. 
For  here  the  circles  with  which  we  compare  angles  are  so  exactly 
similar  that  we  do  not  hesitate  to  pronounce  them  the  same, 
and  even  speak  of  all  circles  as  being  only  one  circle.  And  in 
physical  matters  the  requirement  seems  to  be  satisfied ;  for  our 
physical  world  appears  every  day  to  be  made  up  of  things  so 
exactly  alike  that  we  consider  them  to  be  the  same  things,  and 
although  there  are  some  new  formations  and  destructions  of  old 
things,  yet  a  little  induction  teaches  that  the  matter  in  these,  or 
thk  ultimate  bodies  composing  them,  are  constantly  the  same. 
But  in  economics  the  state  of  things  is  very  different.  From 
age  to  age,  from  century  to  century,  even  from  year  to  year  and 
from  week  to  week,  the  economic  world  is  a  different  world, 
composed  of  many  things  at  one  time  which  do  not  exist  at 
another.  For  our  economic  world  is  only  a  part  of  tlic  whole 
material  world,  and  inay  draw  not  only  new  bodies,  but  new 


THK    TRUE    PIX'ULIARITV  71 

matter  from  it,  and  return  old  tilings  to  it.  In  the  economic 
world  there  is  creation  and  annihilation.  If  particular  things, 
when  consumed,  were  always  replaced  by  similar  things,  and 
nothing  new  were  produced,  we  should  always  have  exactly 
similar  worlds,  which  is  all  we  want.^  But  not  only  the  par- 
ticular things  appear  in  different  quantities,  constituting  classes 
of  different  sizes,  but  wholly  new  classes  come  into  existence  at 
times,  and  some  old  ones  pass  out,  or  qualities  become  better  or 
worse,  really  constituting  different  classes.  And  this  is  not  all. 
At  the  same  time  the  economic  world  in  one  locality  is  different 
from  the  economic  world  in  another. 

Here  we  have  what  probably  constitutes  a  difference  in  kind 
between  the  subjects  of  mensuration  in  economics  and  in  the 
physical  sciences.  It  may  be  that  there  is  no  creation  or  anni- 
hilation in  the  material  universe,  which  therefore  is  the  same 
whole  always ;  wherefore  the  economic  world  differs  from  that 
whole  in  kind.  And  this  difference  in  kind  would  seem  to  go 
over  into  our  mensuration  of  exchange-value  compared  with 
our  mensuration  of  physical  attributes,  such  as  length  and 
weight. 

§  2.  Yet  compared  with  our  practical  measurement  of  phys- 
ical attributes  this  generic  distinction  does  not  exist  in  entirety  ; 
for  we  never  use  the  relation  between  the  parts  and  the  whole 
universe  as  our  standard,  but  only  the  relation  of  the  parts  to 
some  lesser  whole  within  the  larger  whole,  and  this  lesser  whole, 
being  only  a  part  of  the  larger  whole,  may  receive  new  mattei 
from  the  outside  or  yield  up  old  matter  to  the  outside.  For 
instance,  we  measure  the  weight  of  bodies  by  their  relation  to 
the  earth  below  them ;  but  through  volcanoes  the  earth  below 
them  sends  forth  matter  to  the  air  above  them,  and  from  outer 
space  it  receives  meteorites,  so  that  the  whole  with  which  the 
weight  of  bodies  is  compared  is  a  variable  one.     These  changes 

^  For  as  it  is  not  the  other  things  themselves  which  constitute  a  thing's  par- 
ticular exchange-values,  but  these  are  its  power  of  purchasing  the  other  things, 
it  is  inditterent  what  the  other  things  are,  provided  they  be  alike.  The  difficulty 
does  not  lie  in  the  distinctness  of  the  tilings,  but  in  their  differences.  The  prin- 
ciple is  broad.  We  do  not  know  that  tlie  physical  things  we  see  every  day  arc  the 
same  ;  but  this  ignorance  is  no  source  of  trouble  to  the  physicist. 


72  MEASUREMKN'T    OF    (JENEHAE    EXCHANGE-VAEr E 

are  relatively  so  small  that  their  influence  is  imperceptible  to 
us,  and  we  neglect  them.  It  is  worth  enquiring,  however, 
what  \vc  should  do  if  they  were  large  enough  to  provoke  at- 
tention. 

It  is  well  known  that  the  meter  is  supposed  to  be  a  definite 
portion  of  the  circumference  of  the  earth.  Now  if  a  small 
planet,  or  comet,  were  to  collide  with  the  earth  and  unite  itself, 
to  it,  perceptibly  enlarging  our  world,  and  some  of  us  should 
survive  the  catastrophe,  it  is  probable  that,  as  already  remarked, 
Ave  should  regard  the  same  meter  to  be,  not  the  same  proportion 
to  the  new  earth,  but  the  same  proportion  to  that  part  of  the 
new  earth  wliich  alone  constituted  the  old  earfh.  Or  if  the 
planet  in  colliding  with  our  earth  should  scoop  away  some  of  it, 
and  carry  it  oiF  into  space,  we  should  want  our  meter  to  remain 
the  same  proportionately  to  the  smaller  earth  as  it  was  before 
to  this  same  part  of  the  old  earth.  In  other  words,  we  should 
disregard  accretions  and  subtractions,  and  use  for  our  constant 
whole  with  which  we  compare  the  parts  only  a  whole  which 
exists  at  both  periods  compared — a  whole  common  to  both  the 
periods. 

§  3.  In  economic  matters  it  is  easy  to  imitate  this  procedure. 
There  are  even  two  ways  of  doing  so.  The  one  is  this.  In 
measuring  the  exchange-value  of  money  at  two  periods  when 
the  material  constitution  of  the  economic  world  has  varied,  we 
might  merely  take,  in  every  class  of  goods,  the  largest  amount 
of  it  which  exists  at  both  periods.  Thus  if  one  class  has  grown 
larger,  we  should  cut  off  Avhat  has  been  added  at  the  second 
period,  and  take  into  account  only  the  quantity  of  the  first 
period.  Or  if  another  class  has  diminished,  we  should  disre- 
gard the  surplus  which  existed  at  the  first  period,  and  take 
only  the  quantity  of  the  second  period.  The  world  so  reduced 
would  be  a  world  which  exists  at  both  tlie  periods.  And  we 
might  do  the  same  if  we  were  com})arlng  two  economic  worlds 
at  the  same  time,  but  in  different  places. 

The  other  way  is  this.  Instead  of  taking  what  is  common 
to  both  periods  in  every  class  separately,  we  might  take  what 
is  common  to  both  the  |)eri()ds  in  all  the  classes  together  as  a 


THE   TllUK    PEfrLIArvrTY  73 

Avhole.  Thus  in  measuring  the  exchange-value  of  money  in 
commodities  we  should  comj)are  the  total  mass-quantities  of 
goods  a  given  sum  of  money  will  purchase  at  each  period  in 
the  proportions  in  which  the  total  sum  of  money  is  found  to 
have  been  spent  on  the  goods  at  each  period.  Or,  reversely, 
in  comparing  the  exchange-values,  collectively,  of  all  commodi- 
ties in  money,  we  should  compare  the  total  sums  of  money  a 
given  mass-quantity  of  goods  Avill  command  at  each  period, 
this  mass-quantity  l)eing  composed  at  each  period  of  classes  in 
the  proportions  in  which  the  actual  mass-quantity  of  goods  is 
found  to  have  been  composed  at  each  period.  In  order  to  carry 
out  this  method,  we  shall  need  to  find  what  constitutes  same- 
ness in  a  mass-quantity  of  goods  at  different  periods,  or  in  dif- 
ferent places. 

The  second  of  these  methods  is  conformable  to  the  analogy 
of  physical  mensurations.  In  physical  matters  our  standard 
whole  does  not  have  to  be  composed  of  classes  of  things  indi- 
vidually the  same  or  similar  at  both  the  periods  compared. 
This  is  plainer  in  a  more  complex  case.  Suppose  it  should 
miraculously  (as  we  should  say)  happen  that  some  classes  of 
physical  things  should  be  enlarged  by  creation  of  new  bodies 
and  others  diminished  by  annihilation  of  old  ones.  Then  we 
should  want  to  eliminate  only  the  excess  in  the  total  of  the  one 
period  over  the  other.  What  we  want  in  our  standard  Avhole 
is  that  it  should  at  all  periods  be  composed  of  the  same  amount 
of  substance  or  material.  !N^ow  in  economic  things  the  substance 
or  material  is  utility — or  the  importance  we  attach  to  things. 
Hence,  we  want  our  whole  at  each  period  to  contain  the  same 
amount  of  utility,  or  importance  ;  and  it  is  indifferent  how  this 
is  distributed  in  the  various  classes  of  things. 

It  would  seem,  then,  that  even  this  difficulty,  which  threat- 
ened to  be  disastrous,  may  be  overcome.  The  analogy  of  the 
mensuration  of  exchange- value  with  other  kinds  of  mensura- 
tion may  again  be  made  perfect.  The  oidy  ])eculiarity  that  re- 
mains in  the  mensuration  of  exchange-value  is  that  here  the 
wholes  given  us  by  nature  are  not  the  same  at  both  periods,  so 
that   we  have   the  task  of   reducing  them  ;   while   in    j)hysical 


74  MEASUREMENT    OF    GEXEEAT.    EXCHAXGE-VALUE 

subjects  the  material  world  seems  to  be  given  to  us  l)y  nature 
as  a  constant — or  at  least  as  a  sufficiently  close  approximation 
to  a  constant.  The  economic  world  may  be  admitted  to  be  dif- 
ferent in  kind  from  the  piiysical ;  yet  economic  mensuration  is 
the  same  as  the  physical. 

A  certain  defect  must  further  exist  in  measuring  the  general 
exchange-value  of  anything.  This  is  the  impossibility  of  taking 
into  account  all  the  things  which  possess  exchange-value  even 
in  ordinarily  small  economic  worlds  during  even  very  short 
periods.  There  is  a  necessary  confinement  of  our  attention  to 
the  exchange-value  of  money  in  the  most  prominent  classes  of 
staple  commodities.  But,  again,  this  defect  is  not  peculiar  to 
the  mensuration  of  exchange-value ;  for  in  all  mensuration  we 
omit  notice  of  a  major  part  of  the  universe,  and  generally  our 
standards  are  relationships  to  lesser  wholes  within  the  complete 
whole. 

§  4.  It  happens,  however,  that  we  do  not  need  so  much  pre- 
cision in  the  mensuration  of  exchange- value  as  we  do  in  the 
mensuration  of  many  other  quantitative  attributes.  We  need 
precision  in  any  measurement  only  to  the  extent  necessary  to 
preV'Cnt  the  discovery  of  misfits  in  any  subsequent  combinations. 
Thus  if  a  surveyor,  measuring  round  a  field,  reckons  that  his 
two  last  points  are  a  certain  distance  apart,  and  then  finds 
that  they  are  not  that  distance  apart,  his  first  measurements 
have  not  been  conducted  with  the  accuracy  desirable,  and 
corrections  must  be  made.  But  in  the  measurement  of  the  ex- 
change-value of  money  over  several  periods  there  are  few  op- 
portunities for  the  exercise  of  correction  more  sure  than  the 
original  calculation,  if  conducted  on  right  principles  and  with 
care.  At  least  this  is  the  condition  to-day  ;  for  there  is  no 
knowing  what  degrees  of  accuracy  may  in  future  be  obtain- 
able, and  tiierefore  be  desired.  All  that  we  need  at  })resent  to 
strive  after  is  to  find  the  proper  method  whereby  we  may 
rectify  the  carelessly  made  calculations,  or  guesses,  which  every- 
body is  apt  to  make — some  asserting  that  money  has  appreci- 
ated, much  or  little,  others  that  it  has  depreciated,  others  again 
that  it  has  not  varied  at  all, — and  to  approximate  as  nearly  as 


THE    TRUE    PECULIARITY  75 

possible  to  the  truth,  thereby  reaching  a  result  which  nobody 
can  question/ 

2  J.  B.  .Say  made  the  statement  that  in  economics  the  problem  of  finding  a 
constant  exchangu-valiie  is  lilce  the  problem,  in  geometry,  of  squaring  the  circle, 
in  that  both  are  unattainable  with  perfect  exactness,  op.  cit..  Vol.  II.,  p.  89,  cf. 
Cours  complefd'ecoiioniic  politique  pratique,  2d  et].,  p.  181.  This  statement  has 
often  been  repeated  as  if  it  expressed  the  hopelessness  of  all  attempts  to  measure 
exchange-value.  Yet  if  we  could  attain  to  anything  like  the  approximation  to 
exactness  in  measuring  exchange-value  between  two  periods — or  in  measuring 
anything  else — that  we  can  reach  in  measuring  the  ratio  of  the  circumference  to 
the  diameter,  we  should  have  verj'  good  reason  to  congratulate  ourselves. — Mac- 
leod  has  altered  the  simile  by  saying  the  search  after  an  invariable  standard  of 
value  is  like  the  search  after  the  philosopher's  stone  or  perpetual  motion,  7?/c- 
ments  of  political  economy,  1858,  p.  171.  It  is  strange  that  what  is  not  only  a 
legitimate  but  a  necessary  problem  in  economic  science  should  by  an  economist 
be  likened  to  things  which  never  were  objects  of  science,  but  only  of  cupidity. 


CHAPTER   lY. 

SELECTION  AND  ARRANGEMENT  OF  PARTICULAR 
EXCHANGE- VALUES. 


§  1.  In  order  to  compare  the  general  exchange- value  of  any 
one  thing  (generally  money)  at  different  periods,  a  preliminary 
labor  is  that  of  obtaining  an  expression  for  its  general  exchange- 
value  at  each  period  separately.  To  do  this  involves  two  dis- 
tinct operations.  The  one  is  to  select  and  properly  arrange 
the  thing's  particular  exchange-values.  The  other  is  to  com- 
bine these  into  the  thing's  one  general  exchange-value,  which 
they  compose.  The  latter  has  been  the  subject  of  more  dispute 
than  the  former,  which,  though  also  a  subject  of  many  discord- 
ant opinions,  has  not  received  the  attention  it  deserves,  and  is 
by  no  means  so  easy  a  problem  as  it  has  been  taken  for.  It  is 
the  subject  which  naturally  calls  for  attention  first,  although 
it  may  not  admit  of  complete  solution  independently  of  the 
other. 

To  obtain  with  complete  theoretical  exactness  the  exchange- 
value  of  money — or,  to  be  precise,  of  a  certain  sum  of  money, 
say  the  money-unit — at  any  place  during  any  period — a  week, 
a  month,  a  year, — we  ought  to  take  account  of  every  individual 
thing  which  has  been  exchanged  at  that  place  during  that 
period,  and  of  its  price  or  exchange-value  in  money,  which  will 
give  the  money-unit's  exchange-value  in  it,  whether  it  was 
actually  exchanged  for  money  or  not.  To  do  this  is  impossible  ; 
and  so  our  practical  measurement  of  general  exchange-value 
during  any  period  is  subject  to  curtailment.  The  whole  which 
we  can  employ  can  only  be  a  part  of  the  total  of  exchangeable 

76 


EXCTATSIONS    AND    INCLUSIONS  77 

things.  Bat  our  efforts  must  be  directed  at  making  this  prac- 
ticable whole  as  large  as  possible  ;  for,  as  has  been  said  by  an 
eminent  investigator  in  this  subject,  "  the  result  is  more  accu- 
rate, the  greater  the  number  of  the  data,  and  the  smaller  the 
number  of  omitted  articles."  ^ 

§  2.  The  curtailment  must  begin  by  leaving  out  of  account 
things  which  appear  only  as  individuals — such  as  race  horses, 
paintings,  antiques  and  the  like.  These  individual  things  must 
be  omitted  not  only  because  their  number  is  legion,  but  because 
each  one  is  exchanged  only  occasionally,  and  so  would  not  ap- 
pear in  all  the  successive  periods,  and  none  can  stand  for  an- 
other. Moreover  their  omission  is  only  a  small  loss,  as,  in  spite 
of  their  great  numbers,  their  sales  all  told  form  but  a  small  part 
of  the  immense  quantity  of  all  sales. 

The  majority  of  exchangeable  things  fall  into  classes,  under 
generic  or  specific  names,  in  w^iich  all  individuals  at  the  same 
time  and  place  have  the  same  price,  or  different  prices  accord- 
ing to  different  qualities,  which  form  sub-classes  ;  and  there  is 
always  a  succession  of  similar  individuals  appearing  in  the 
market  during  every  period.  These  classes  we  can  employ  in- 
stead of  the  individual  things  themselves,  and  so  our  labor  is 
already  enormously  reduced.  For  when  one  of  the  classes, 
representing  the  individuals  in  it,  is  said  to  vary  in  exchange- 
value  or  in  price,  we  at  once  know  that  all  the  individuals  in  it 
have  so  varied.  The  quoted  price  of  a  bushel  of  wheat  is  not 
the  price  of  a  particular  bushel  of  wheat,  but  of  any  bushel  of 
wheat  (of  the  same  quality)  at  the  same  time  and  place. 

All  classes  of  things,  hoAvever,  do  not  equally  well  represent 
the  individuals  in  them  ;  for  in  some  classes  the  individuals 
vary  infinitely  in  quality  or  in  size  or  in  many  attributes  that 
go  to  make  them  valuable.  Thus  all  complex  products,  such  as 
machines,  buildings,  ships,  railroads,  are  too  variegated  in  their 
individuals  to  make  the  logical  subsumption  of  these  into  classes 
of  much  importance  for  the  object  we  have  in  view.  Therefore 
these  things,  as  also  fancy  breeds  of  animals,  precious  stones, 
and  most  articles  of  luxury,  must  be  neglected  as  being  little 

1  Edgeworth,  B.  60,  p.  197. 


78  SELECTION    OF    PARTICULAR    EXCH A  ^'GE- VALUES 

more  than  collections  of  untractable  individuals.  The  omission 
of  coijiplex  products  is  of  little  consequence,  because  their  ex- 
change-values generally  vary  in  somewhat  the  same  way  as  do 
the  materials  of  which  they  are  made  or  the  simple  products 
which  they  help  to  make.  Land  -also  is  such  a  heterogeneous 
class,  embracing  lots  and  fields  and  forest  districts  of  infinitely 
various  exchange-values  according  to  situation  and  natural  fer- 
tility. The  omission  of  these  from  an  attempt  to  measure  the 
general  exchange-value  of  money  in  all  exchangeable  things  is 
of  considerable  moment.  But  if  we  are  seeking  rather  to  es- 
tablish for  the  exchange-value  of  money  a  standard  composed 
of  products,  land  would  not  belong  to  this.  Stocks  or  shares 
in  railroads  and  industrial  companies  are  not  to  be  counted, 
partly  because  the  prices,  or  money  costs,  of  these  are  not  to  be 
counted  for  the  reason  just  given,  and  partly  because  their 
prices  are  dependent  upon  the  profits,  wliich  are  greatly  depend- 
ent upon  the  prices  of  the  services  or  products,  already  counted. 
For  a  similar  reason  the  money  cost  of  transportation  of  goods 
ought  not  to  be  counted,  because  it  is  a  factor  in  the  price  of 
goods,  and  so  is  already  counted  in  them.  But  the  money  cost 
of  transportation  of  persons,  or  of  travel,  ought  to  be  counted. 
Bonds  are  not  to  be  counted  because  their  prices  depend  upon 
the  general  rate  of  interest  on  the  one  hand  and  on  the  other 
upon  the  particular  credit  of  each  company,  both  of  which  fac- 
tors have  nothing  to  do  with  the  hiake-up  of  the  exchange- 
value  of  money,  although  they  are  both  aifected  by  variations 
in  the  exchange-value  of  money,  which  therefore  needs  to  be 
measured  independently  of  them. 

The  classes  to  be  counted  are,  then,  to  be  confined  to  so-called 
raw  products  and  to  those  things  which  have  been  called  fungi- 
ble, namely  things  sufficiently  alike,  every  one  in  its  own  class, 
to  replace  one  another  with  indifference  on  the  part  of  their 
owners,  which  are  things  that  can  be  meted  out  by  weight  or 
other  measures,  or  by  the  piece.  These  things  include  not  only 
raw  products — grain,  cattle,  metals,  etc. — but  also  many  manu- 
factured products  in  a  medium  stage — steel,  flour,  yarn,  cloth 
— and  a  few  finished  products — rope,  some  simple  tools,  bread, 


EXCLUSIONS    AND    INCIATSIONS  79 

and  even  ready-made  elothing  and  shoes,  which  are  now  turned 
out  in  large  (juantities  in  ahnost  imiform  grades.  All  of  these 
classes,  which  continuously  provide  a  succession  of  very  nearly 
similar  individuals  definitely  measured  and  priced,  are  to  be 
sought  for  and  included.  Perhaps  two  or  three  hundred  such 
classes  may  be  found,  which  number  is  not  too  large  to  be  dealt 
with  by  a  board  of  trained  statisticians,  collecting  data  and 
making  calculations  in  the  course  of  present  time. 

§  3.  In  trying  to  discover  the  ups  and  downs  of  the  exchange- 
value  of  money  in  times  past,  it  is  impossible  to  find  continuous 
price-lists  of  so  many  classes,  or  information  about  the  quanti- 
ties of  the  individual  things  composing  them  ;  so  that  here  we 
must  confine  our  researches  to  a  few.  The  few  then  must  be 
chosen  as  samples.  Now  serviceable  for  samples  are  :  (1)  only 
the  most  important  or  staple  products,  (2)  only  those  things 
whose  prices  are  independent  of  each  other,  (3)  only  those  whose 
prices  are  not  subject  to  special  causes  of  fluctuation  ;  and,  fur- 
thermore, (4)  some  of  these  articles  should  not  appear  twice  in 
the  lists,  once  as  a  raw  material,  and  again  in  the  manufactured 
article  made  of  it,  while  others  appear  only  once.  Some  econo- 
mists have  even  advocated  this  "  exclusive  "  method  for  present 
times.  There  is  no  good  reason  for  doing  so.  This  method  is 
only  a  makeshift,  the  best  we  can  do  in  reviewing  past  times. 
We  can  do  better  for  present  times,  and  should,  therefore,  seek 
to  do  so.^  Some  of  the  early  writers  were  content  to  measure 
the  variations  in  the  exchange-value  of  money  by  comparing  it 
only  with  one  other  article,  generally  wheat,  or  the  most  com- 
monly used  food  product.'  The  line  of  progress  has  been  to 
widen  the  range  of  the  comparison  more  and  more,  until  it 
reaches  the  utmost  limit  practicable.     Now,  it  should  be  noticed, 

^  Slightly  different  is  the  position  of  those  who  advocate  a  "  multiple  standard  " 
for  contracts,  or  for  regulating  the  exchange-value  of  paper  money,  composed  of 
twenty,  of  a  hundred,  or  of  any  other  small  number  of  classes,  selected  for  their 
importance  or  from  other  motives.  Their  position  is,  not  that  this  procedure 
yields  the  best  measure  of  the  exchange-value  of  money,  but  that  it  forms  a  mucii 
better  standard  than  the  standard  resting  on  one  or  two  metals.  Still,  while  they 
are  about  it,  they  might  as  well  have  the  largest  "  multiple  standard  "  possible. 

-  So  Locke,  Adam  Smith,  Condillac,  Say,  D.  Stewart,  Storch,  Cibrario,  and 
others. 


80      AERANGEMENT    OF    PARTICULAR    EXCHANGE-VALUES 

the  canons  suitable  for  the  "  exclusive  "  method  have  little  or 
no  application  for  the  broad  "inclusive"  method.^ 

§  4,  In  all  cases  only  wholesale  prices  are  to  be  employed. 
This  is  because  they  are  the  only  ones  practicable.  Objection 
is  often  made  that  it  is  retail  prices  which  give  the  exchange- 
value  of  money  to  the  consumers.  This  is  not  altogether  true, 
as  the  economic  "consumer"  is  often  the  purchaser  in  whole- 
sale quantities,  as  the  manufacturer,  who  consumes  raw  mate- 
rials ;  and  also  many  large  institutions,  hotels,  governments, 
etc.,  mostly  get  their  stores  at  wholesale  prices.  Although  on 
some  occasions  retail  prices  do  not  follow  the  variations  of 
wholesale  prices,  and  may  even  vary  slightly  while  those  are 
stable,  it  is  probable  that  the  true  average  of  retail  prices  fol- 
lows the  ascertained  average  of  wholesale  prices  much  more 
closely  than  the  attempt  to  reckon  the  average  of  retail  prices 
would  succeed  in  following  the  true  average.* 

II. 

§  1.  We  have  seen  that  when  a  class  of  commodities  varies 
in  price,  all  the  individuals  in  it  do  so.  Now  if  one  class  is 
larger  than  another,  when  they  both  vary  in  price,  the  variation 
of  the  larger  class  represents  a  variation  of  more  individuals, 
and  the  variation  of  money  in  exchange-value  in  that  class  is  a 
variation  of  it  in  more  individuals,  than  is  its  variation  in  the 
other  class.  The  particular  exchange- value  of  money  in  a  larger 
class  is  not  necessarily  a  higher  or  greater  exchange-value,  but 
it  is,  so  to  speak,  a  wider  or  larger  exchange-value,  than  its 
particular  exchange-vajue  in  a  smaller  class.  Therefore  the 
variation  of  money  in  the  larger  class  should  count  for  more, 
or,  which  is  the  same  thing,  the  variation  in  price  of  the  larger 

3  The  distinction  is  not  observed  in  the  followinf?  statements:  "No  article 
should  be  scheduled  twice  in  different  stages  of  manufacture,"  J.  B.  Martin,  Gold 
versus  goods,  in  the  Journal  of  the  British  Association  for  the  Advancement  of 
Science,  for  the  year  1883,  p.  626 ;  "  We  might  count  the  wool  instead  of  the  things 
made  of  it  (for  of  course  wc  ought  not  to  count  both),"  Marshall,  B.  93,  p.  373. 

■*  At  all  events,  if  retail  prices  are  used,  they  must  ))e  used  in  a  measurement 
by  themselves.  Wholesale  and  retail  prices  must  never  be  mixed  in  one  table. 
So  Beaujon,  B.  95,  pp.  llU-111,  114,  116. 


NEED    OF    WEKillTINC  81 

class  should  count  for  more.  The  making  allowance  for  the 
sizes  of  the  classes,  which  consists  in  assigning  to  each  class  its 
proper  importance  or  weight  in  the  calculation  of  the  general 
variation  of  prices  has  been  called  "  weighting,"  and  the  size 
assigned  to  each  class  has  been  called  its  "  weight."  These 
not  very  well  chosen  terms  have  become  consecrated  by  usage.' 
We  should  notice  that  the  idea  of  vcU/ld  attaching  to  all  classes 
refers  to  their  influence  upon  the  result  in  our  calculations. 
The  relative  weights  of  the  classes  in  our  calculations  are  not 
relative  weights  of  the  classes  per  se.  They  are  dependent  upon 
the  relative  sizes  of  the  classes  per  se. 

§  2.  The  prices  of  the  classes  of  things  are  quoted  in  the 
market  lists  on  mass-units  of  various  kinds  and  magnitudes, 
diiferently  in  different  places  and  times,  as  it  has  been  found 
convenient  for  merchants  to  bargain  for  (e.  </.,  iron  by  the  ton 
and  copper  by  the  pound).  The  quoted  prices  of  some  things 
are  therefore  very  much  higher  than  those  of  others,  without 
reference  either  to  the  preciousness  or  to  the  importance  of  the 
classes.  If  these  prices  be  taken  simply  as  they  occur  and  be 
combined  in  a  certain  way,  the  operation,  as  will  be  shown 
later,  is  virtually  that  of  weighting  the  classes  according  to  the 
accidental  height  of  their  quoted  prices  at  the  first  period,  in  a 
comparison  of  two  or  more  periods.  Thus  the  weighting  here 
is  purely  accidental  and  haphazard,  without  any  principle  or 
reason  for  assigning  more  importance  to  one  class  than  to  an- 
other, except  the  chance  of  mercantile  customs,  which  have 
grown  up  without  reference  to  this  subject.  Such  a  method  of 
calculating  general  exchange-value  and  its  variations  was  em- 
ployetr — of  course  without  knowledge  of  what  they  were  doing — 

1  The  terms  have  long  been  in  use  in  the  reduction  of  observations,  especially 
in  astronomy.  They  appear  to  have  been  introduced  into  economics  by  Jevons, 
who  in  his  important  first  work  on  our  subject  wrote  :  "  It  must  be  confessed  that 
the  exact  mode  in  which  preponderance  of  rising  or  falling  prices  ought  to  be  de- 
termined is  involved  in  doubt.  Ought  we  to  take  all  comnjodities  on  an  equal 
footing  in  the  determination  ?  Ought  we  to  give  most  weight  to  those  which  are 
least  intrinsically  variable  in  value?  [Cf.  ^Malthus,  above  in  Chapt.  I.,  Sect.  I., 
Note  8.]  Ought  we  to  give  additional  weight  to  articles  according  to  their  im- 
portance, and  the  total  quantities  bouglit  and  sold?  The  question,  when  fully 
opened,  seems  to  be  one  that  no  writer  has  attempted  to  decide — nor  can  I  attempt 
to  decide  it."  B.  22,  p.  21. 
6 


'82      AURANGEMENT    OF    PARTICULAR    EXCHANGE- VALUES 

by  one  of  the  earliest  investigators  in  this  subject  in  the  eight- 
eenth century,  the  French  financier,  Dutot,  and  in  the  century 
just  elapsed  even  by  two  prominent  economists.^  It  was  avoided 
in  the  middle  and  toward  the  end  of  the  prior  century  by  two 
other  investigators,  the  Italian  writer  on  monetary  matters, 
Carli,  and  the  English  physicist,  Sir  George  Shuckburgh  Evelyn, 
the  latter  of  whom  introduced  the  practice  of  reducing  all  prices 
:at  one  period,  taken  as  basis,  to  100  (/.  e.,  100  money-units), 
■whatever  be  the  quantities  of  goods  which  are  sold  at  this  price 
in  each  class,  or  whatever  be  the  number  of  times  this  money's 
worth  of  goods  of  each  class  are  sold  during  the  periods  com- 
pared. Here,  though  it  is  usual  to  say  there  is  no  weighting, 
there  really  is  weighting,  since  all  the  classes  are  treated  as  if 
they  were  of  the  same  size — there  is  even  weiyhthig,  which  is 
only  next  W'Orst  after  the  earlier  haphazard  weighting.  It  is, 
in  fact,  impossible  to  avoid  assigning  one  weight  or  another  to 
the  classes  reviewed  ;  wherefore  it  is  plain  that  the  proper  thing 
for  us  to  do  is  to  distribute  the  weighting  with  the  greatest  care 
possible. 

§  3.  Not  infrequently  it  has  been  asserted  that  it  is  shown  by 
experience  not  to  be  worth  the  trouble  to  assign  proper  weiglit- 
ing  to  classes,  as  the  so-called  "  unweighted "  calculations 
(really  w^ith  even  weighting)  yield  results  very  little  divergent 
— so  it  is  claimed — from  those  reachf^d  with  proper  uneven 
weighting.  Whether  it  is  worth  the  trouble  to  be  careful,  even 
though  the  divergence  were  small,  would  seem  to  depend  some- 
what upon  the  question  who  is  to  take  the  trouble ;  for  what 
would  excuse  a  voluntary  investigator  surveying  past  events 
for  a  merely  historical  purpose  would  not  excuse  a  board  of  of- 
ficial statisticians  employed  by  the  State  for  the  practical  pur- 
pose of  providing  a  guide  for  contracts  or  for  the  regulation 
of  money  in  the  present  course  of  time.  But,  although  the  gen- 
eral trend  upward  or  downward  in  a  series  of  years  may  be 
somewhat  similar  in  the  two  methods,  the  particular  results  are 
often  very  divergent,  as  shown  by  the  very  calculations  upon 

-  See  Appendix  U,  I.  As  an  example  of  the  curious  combinations  this  method, 
or  want  of  method,  may  lead  to,  it  maybe  noticed  [that  in  one  of  his  calculations 
Levasseur  allowed  HHO  times  greater  influence  to  tin  than  to  cotton  ! 


NEED    OF    WEIGHTING  83 

which  the  other  opinion  has  been  fonnded.  Thus  in  the  com- 
parison given  by  Mr.  Palgrave  of  the  Econoiaist  series  of  "  un- 
weighted" index-numbers  and  tlie  "weighted"  index-numbers 
calculated  upon  the  same  prices,  we  find  the  following  contrasts  : 


Year 

Evenly  weighted 

Unevenly  weighted 

1880 

"  87 

89 

1881 

81 

93 

1882 

83 

87 

1883 

79 

88" 

Here  the  calculated  movements  of  general  prices  go  in  ex- 
actly opposite  directions  in  every  sequence  of  years.  Between 
the  first  and  the  second  vears,  for  instance,  the  Economist  fio-ure 
falls  7  per  cent.,  and  the  "  corrected  "  figure  rises  4|  per  cent., 
— a  diiference  of  12  per  cent.  Divergences  of  this  sort  are  to 
be  seen  in  every  case  where  in  a  series  of  periods  the  same 
prices  have  been  treated  in  both  ways  for  comparison.  By  the 
same  argument,  therefore,  by  which  it  has  been  attempted  to 
show  the  needlessness  of  uneven  weighting,  the  need  of  it  is 
proved.^ 

To  assign  the  "  weights  "  with  perfect  precision  would  involve 
a  great  amount  of  labor — principally  in  discovering  the  relative 
sizes  of  the  classes  ;  for  the  mere  introduction  of  the  ascertained 
weights  in  the  calculations  does  not  much  increase  the  labor. 
But  to  assign  uneven  weighting  with  approximation  to  the  rela- 
tive sizes,  either  over  a  long  series  of  years  or  for  every  period 
separately,  would  not  require  much  additional  trouble ;  and 
even  a  rough  procedure  of  this  sort  would  yield  results  far 
superior  to  those  yielded  by  even  weighting.  It  is  especially 
absurd  to  refrain  from  using  roughly  reckoned  uneven  weight- 
ing on  the  ground  that  it  is  not  accurate,  and  instead  to  use 
even  weighting,  which  is  much  more  inaccurate. 

3  B.  77,  pp.  329-330. 

*  We  shall,  however,  hiter  find  that  both  the  methods  used  in  the  above  cal- 
culations are  very  defective,  and  especially  so  in  the  matter  of  afibrdins  com- 
parison between  any  two  years  neither  of  which  is  the  basic  year,  so  that  much  of 
their  divergences  maybe  due  to  error  even  in  the  "corrected"  figures.  What 
difference  would,  then,  exist  in  practice  between  the  true  method  and  the  simple 
method  is  still  an  unknown  quantity.  But  we  shall  later  also  find  that  in  the 
proper  way  of  forming  a  series  of  index  numbers  a  slight  error  in  every  calculation 
may,  in  a  wrong  method,  accumulate  before  long  into  large  error.  Therefore 
every  contrivance  to  secure  accuracy  is  imperative. 


84         ARRANGEMENT    OF    PARTICULAR    EXCH A N(;E- VALUES 

§  4.  Still,  although  few  of  the  practical  investigators  have 
actually  employed  anything  but  even  weighting,  they  have 
almost  always  recognized  the  theoretical  need  of  allowing 
for  the  relative  importance  of  the  different  classes  ever  since 
this  need  was  first  pointed  out,  near  the  commencement  of  the 
century  just  ended,  by  Arthur  Young/  But  the  method  of 
measuring  the  sizes  of  the  classes  has  been  the  subject  of  diverse 
views,  and  even  the  reasons  offered  for  the  most  commonly 
adopted  method  have  not  been  quite  satisfactory. 

Arthur  Young  advised  ,  simply  that  the  classes  should  be 
weighted  according  to  their  importance.  Some  early  critics  of 
the  plan  of  judging  the  value  of  money  "  by  its  relation  to 
the  mass  of  commodities "  objected  that  the  importance  of 
articles  is  different  to  different  persons,  and  therefore  there 
could  be  no  one  standard  of  this  sort."  Joseph  Lowe  yielded 
so  much  to  this  criticism  as  to  recommend  for  different  ranks  of 
society  different  weightings,  and  consequently  so  many  different 
standards  ;  but  again  he  wanted  one  standard  to  be  formed  with 
weighting  according  to  the  importance  of  the  classes  for  the 
whole  community,  as  indicated  by  the  total  "  values "  con- 
sumed.^ That  the  classes  should  be  weighted  according  to  their 
relative  total  money-values,  has  become  the  prevalent  doctrine, 
being  frequently  re-invented  as  if  by  instinct,^  and  generally 
with  the  same  explanation.     One  of  the  best  forms  in  which 

°  Young  did  so  in  opposition  to  Evelyn's  method,  which  lie  condouined  as 
manifestly  wrong  in  counting  the  articles  as  equally  important,  B.  6,  pp.  68,  70. 
There  have,  however,  been  relapses.  Thus  Porter,  in  B.  11,  totally  ignored  un- 
even weighting,  although  his  attention  had  been  called  to  it  by  Tooke,  who  there- 
upon declared  his  table  to  be  misleading  because  it  allowed  equal  influence  in  the 
result  to  unimportant  as  to  important  articles.  Evidence  before  the  Select  Com- 
mittee on  Banks  of  Issue,  1840,  q.  8(>15.  Then  a  period  followed  in  which  even 
weighting  was  employed  with  little  notice  of  the  need  of  anything  else,  until 
Dro])isch  demolished  Laspeyres'  method  for  this  neglect,  B.  30,  p.  145,  cf.  B.  20, 
pp.  .32-33.  Laspeyres  admitted  the  theoretical  need  of  uneven  weighting,  B.  2(i, 
p.  304 ;  and  since  then  attention  has  been  paid  to  it  by  almost  all  writers. 

"Ricardo,  Works,  p.  401  ;  ]\[althus,  op.  cit.,  pp.  119-120.  So  also  Von  Jacob, 
the  German  translator  of  Lowe,  according  to  S.  D.  Horton,  Silver  and  gold,  Cin- 
cinnati, 1876,  2d  ed.,  181)5,  p.  3!t. 

■^  B.  8,  Appendix,  pp.  !i3-99. 

8  Tooke,  loc.  cit. ;  Giffen,  B.  45,  q.  709  ;  Sidgwick,  B.  r^G,  p.  66  ;  Palgrave,  B. 
77,  p.  344 ;  Sauerbeck,  B.  79,  p.  595 ;  Marshall,  B.  93,  p.  372  ;  "Wasserab,  B.  105, 
pp.  87,  89,  94  iV.  ;  Westergaard,  B.  110,  p.  220;  I-\)nda,  B.  127,  p.  160. 


XKEl)    OF    WEIGHTING  85 

this  explanation  has  been  presented  is  that  the  wei^liting  then 
follows  the  importance  of  the  articles  to  the  "  average  con- 
sumer." ^  The  standard  of  exchange-value  now  becomes  what 
has  aptly  been  called  a  "  standard  of  desiderata."  "* 

To  this  view  the  objection  has  sometimes  been  made  that  it 
is  too  "  subjective."  An  "  objective  "  criterion  has  been  in- 
voked by  saying  that  "  more  weight  should  be  assigned  to  those 
commodities  which,  being  circulated  in  greater  quantities,  make 
greater  demand  on  the  currency."  "  Here  the  measurement  of 
the  exchange-value  of  money  is  viewed  in  a  peculiar  manner. 
For  no  one  would  dream  of  measuring  the  exchange-value 
of  any  commodity  by  weighting  the  other  articles  in  which 
its  particular  exchange-values  vary  merely  by  the  quantities 
of  them  actually  exchanged  for  that  commodity.  In  order 
that  one  thing  have  exchange-value  in  another  it  is  not  neces- 
sary that  the  two  should  be  actually  exchanged  for  each  other  ; 
and  the  consideration  by  which  the  size  of  the  other  thing's 
class  is  to  be  judged  cannot  possibly  be  confined  to  the  quantity 
of  it  exchanged  for  this  thing.  There  is  nothing  peculiar  here 
in  the  case  of  money.  We  are  not  measuring  the  demand  for 
money,  but  the  general  exchange-value  of  money.  This  doc- 
trine, therefore,  is  not  so  good  as  the  one  it  seeks  to  supersede. 

In  regard  to  the  question  of  objectivity  and  subjectivity,  it 
may  be  noticed  that  these  terms  are  used  in  two  senses.  One 
of  them  is  that  "  objective  "  refers  to  what  is  universal  to  all 

^Marshall,  B.  03,  p.  372. — The  "average  consumer,"of  course,  is  tlie  whole 
community,  and  not  a  few  samples  of  "  normal  families."  The  method  of  aseer- 
taining  the  "average  consumer"  by  means  of  family  budgets  (especially  if  con- 
fined to  those  of  day  laborers,  as  often  done — and  advised  by  Pomeroy,  B.  135,  p. 
332)  is  certainly  not  so  accurate  as  the  method  seeking  the  total  volume  of  goods 
in  trade.  The  investigation  of  family  budgets,  introduced  by  Eden  and  elabo- 
rated by  Engel,  is  very  important  for  sociological  studies,  but  hardly  interests  us 
here.  The  statistician  who  has  used  them  most  in  our  department  is  Falkner  (see 
especially  B.  Ill,  pp.  XL-LV),  who  defends  them  in  B.  112,  pp.  (33-(j4  and  B.  113, 
pp.  269-270.     The  two  methods  are  discussed  by  Taussig,  B.  121,  pp.  24-25. 

1"  Horton,  op.  cit.,  p.  35  ;  The  silver  pound,  1887,  pp.  3-4.  That  such  a  standard 
is  one  of  the  objects  of  these  enquiries  has  been  asserted  by  the  Britisii  Association 
Committee,  First  Report,  B.  99,  p.  249  n. 

11  Edgeworth,  B.  OG,  p.  139.  This  would  lead  to  a  slight  change  in  the  weight- 
ing, as  perceived  by  Foxwell,  who,  according  to  Edgeworth,  B.  (33,  p.  135,  wanted 
weighting  to  be  assigned  according  to  the  values  of  the  classes  multiplied  by  the 
numbers  of  times  the  articles  in  them  are  sold  before  being  consumed. 


8()       ARRANGEMENT    OF    PARTICULAR    EXCHANGE-VALUES 

men,  in  distinction  from  what  is  peculiar  to  individual  men  as 
"  subjective."  In  this  sense  the  above  measurement  of  the 
sizes  of  the  classes  according  to  their  importance  for  the  whole 
community  is  objective.  Subjective  would  be  the  many  stand- 
ards, one  for  each  person  according  to  the  importance  of  the 
classes  to  him  individually.  Still  we  desire  a  measurement,  and 
a  reason  for  it,  which  are  "  objective "  in  the  other  sense  of 
being  out  of  dependence  upon  persons.  The  case  is  somewhat 
like  the  measurement  of  heat.  Heat  is  subjective,  regarded  as 
a  feeling  in  men,  though  it  be  in  all  men.  Its  variations  are 
also  subjective  as  being  diflPerently  felt  by  diiferent  persons. 
But  we  measure  heat  without  regard  to  the  susceptibilities  of 
individual  persons.  And  more  still  :  we  measure  it  without 
regard  to  its  universal  influence  upon  all  men.  A  being  who 
never  had  the  sensation  of  heat,  could  measure  heat.  Could 
not  a  being  who  knows  not  the  idea  of  importance  rightly 
measure  out  the  proper  weighting  to  be  assigned  to  the  classes 
of  exchangeable  articles  ? 

§  5.  A  few  economists  have  held  that  weighting  should  be 
according  to  the  relative  masses  that  are  consumed  of  the  differ- 
ent classes,  all  the  masses  being  measured  l)y  the  same  common 
mass-unit,  a  unit  of  weight  being  generally  preferred.^"  This 
position  claims  to  be  the  truly  objective.  It  has  the  fault  that 
it  may  be  objective  in  two  distinct  ways,  neither  of  which  has 
any  superiority  over  the  other.  For  if  we  measured  the  masses 
of  the  classes  by  a  common  measure  of  capacity,  the  relative 
sizes  of  the  classes  would  turn  out  differently.  It  may  be  re- 
plied that  it  is  weight  Avhich  gives  the  mass  proper,  or  quantity 
of  matter,  contained  in  commodities.  The  counter-reply  is  that 
capacity  is  the  measure  of  volume,  or  size  in  space  ;  and  there 
is  no  reason  why  the  quantity  of  an  inscrutable  thing  called 
matter  should  be  more  considered,  in  economics,  than  the  (pian- 
tity  of  visible  space  which  the  objects  occupy.     We  compare 

12  So  Drobisch,  B.  29,  pp.  30,  35,  B.  30,  pp.  148, 153 ;  Lehr,  B.  68,  p.  11 ;  Lind- 
say, B.  114,  p.  26;  and  apparently  the  British  Association  Committee,  First  Re- 
port, B.  9!),  pp.  249-250:  and  Edgeworth,  B.  59,  p.  264.  But  statements  on  this 
matter  are  often  unprecise,  the  amhisuous  term  "  (juantities  "  being  used.  We 
shall  also  see  that  in  some  cases  weighting  which  appears  to  be  by  masses  is  really 
by  total  money-values.  , 


WEIGHTIN(;    EXTLAINED  87 

some  exchangeable  objects  by  vohime  or  bulk,  and  consider  it 
an  advantage  that  they  be  heavy.  AVe  compare  other  ex- 
changeable objects  by  weight,  and  consider  it  an  advantage  that 
they  be  large  (and  therefore  light).  We  never  compare  all 
things  either  by  weight  alone  or  by  bulk  alone.  And  further- 
more, some  exchangeable  objects,  such  as  land,  have  neither 
weight  nor  volume,  while  other  things,  such  as  railroad  tickets, 
no  one  would  think  of  sizing  according  to  their  weight  or  vol- 
ume ;  and  some  gases,  wliich  now  are  commercial  articles,  and 
sold  by  capacity  at  a  certain  pressure,  it  would  be  difficult  to 
bring  into  line  with  solid  or  liquid  things,  since  the  pressure 
chosen  is  arbitrary.  But  to  this  position  a  more  direct  objec- 
tion will  be  noticed  presently. 

III. 

§  1.  Such  is  a  brief  history  of  the  question  of  weighting 
brought  down  to  the  present  day.  It  is  obvious  that  the  ques- 
tion has  not  been  thoroughly  discussed.  Even  the  nature  of 
weighting  in  general  has  rarely  been  understood,  the  term  it- 
self being  misleading  ;  and  the  special  difficulties  concerning 
weighting  in  our  subject  have  never  been  pointed  out,  wdiere- 
fore  they  have  never  been  overcome. 

We  shall  see  that  anotlier  question  which  naturally  arises  *in 
our  subject  is  a  question  of  averaging  the  variations  of  exchange- 
values.  Xow  weighting  is  a  question  connected  with  averag- 
ing, and  though  even  weighting  is  the  more  common  in  simple 
theoretical  problems,  in  practical  problems  occasion  for  uneven 
weighting  is  always  likely  to  present  itself.  The  question  of 
weighting  must  be  treated  first,  because  weighting  of  some  sort 
is  a  prerequisite  in  all  averaging. 

The  nature  of  weighting  may  be  illustrated  by  a -simple  ex- 
ample. We  may  suppose  that  two  proud  fathers,  each  with 
three  sons,  dispute  as  to  which  has  the  taller  sons,  and  proceed 
to  measure  and  to  average  them.  The  one  finds  a  different 
figure  for  the  tallness  of  each  son.  He  simply  adds  up  the 
three  figures,  and  divides  by  three.      He  has  used  even  weight- 


88       ARRANGE>rEXT    OF    I'AHTICULAFv    EXCHAXCIE-VAT.UF:s 

ing,  allowing  ecjual  importance  Id  the  calculation  to  each  meas- 
urement. The  other,  let  us  suppose,  finds  two  of  his  sons  to 
be  equally  tall  and  the  other  differently  tall.  He  then  has  only 
two  different  figures.  The  one  represents  the  tallness  of  two 
sons,  the  other  the  tallness  of  one  sou.  If  he  adds  these  two 
figures  and  divides  by  two,  the  result  would  be  wrong.  If  he 
notes  down  the  figures  for  every  son  separately,  although  the 
same  figure  occurs  twice,  and  adds  these  three  figures  and  di- 
vides by  three,  he  would  get  the  right  result,  and  be  doing  ex- 
actly what  the  other  did — he  would  be  using  even  weighting, 
with  three  figures.  But  if  he  employs  only  the  two  distinct 
figures,  but  takes  the  one  representing  the  two  measurements 
twice,  by  multiplying  it  by  two,  and  adds  this  doubled  figure 
to  the  other  single  figure,  and  divides  by  three,  he  would  be 
using  uneven  weighting;  for  he  would  be  allowing  twice  as 
mnch  influence  upon  the  result  to  the  one  figure  as  to  the  other. 
The  result  is  the  same  as  if  he  employed  even  weighting  with 
addition  of  three  figures.  In  this  simple  example  the  differ- 
ence in  the  labor  of  performing  either  of  these  operations  is 
not  great.  But  in  more  complex  matters  it  may  amount  to  a 
great  deal.  The  operation  of  first  multiplying  all  the  similar 
figures  by  the  number  of  times  they  occur  and  adding  their 
products  to  one  another  and  to  the  figures  that  occur  only  once, 
and  dividing  by  the  total  number  of  repeated  and  single  figures, 
is  a  simpler  one  than  that  of  adding  all  the  individual  figures. 
Thus  uneven  weighting,  though  appearing  to  be  more  difficult 
than  even  weighting,  is  really  a  means  of  simplifying  and 
abridging  the  calculation. 

Weighting,  then,  consists  in  allowing  for  the  number  of  in- 
dividuals which  possess  a  common  attribute  in  certain  quantities, 
which  are  being  averaged.  Even  weighting  is  employed  when 
every  individual  is  measured,  and  its  measurement,  whether  the 
same  or  different  from  others,  is  separately  employed  in  the  cal- 
culation. Uneven  weighting  is  employed  when  equal  measure- 
ments are  treated  as  one,  for  convenience  merely,  and  yet  allow- 
ance is  made  for  the  number  of  times  they  are  rej)eated.  Or 
uneven  weighting  is  to  be  employed,  again,  when  we  make  only 


WEKJUTINC;    EXPLAINED  89 

a  few  observations  on  different  classes  of  individuals,  and  know 
(or  assume)  that  the  other  individuals  are  like  those  measured  ; 
for  then  the  quantity  of  the  attribute  possessed  by  each  indi- 
vidual in  the  class  is  really  to  be  repeated  the  number  of  times 
it  is  possessed  by  an  individual,  that  is,  by  the  number  of  in- 
dividuals in  the  class.  Only  if  the  numbers  of  individuals  in 
all  the  classes  happen  to  be  the  same,  can  even  weighting  be 
properly  employed. 

§  2.  Now  in  most  subjects  of  averaging,  the  nature  of  the 
individuals  in  each  class  is  directly  given,  wherefore  the  finding 
of  their  numbers  is  not  difficult — at  least  theoretically  speaking. 
Thus,  in  the  above  example,  the  three  sons  in  each  set  were  the 
individuals  in  question.  Or  in  the  wider  case  of  measuring  the 
average  tallness  of  the  population  of  a  country,  after  getting 
the  average  tallness  for  different  districts,  we  should  weight  the 
calculation  of  the  general  average  of  these  by  the  numbers  of 
persons  in  each  district.  This  is  because,  by  the  nature  of  the 
problem,  every  full-grown  person,  be  he  prince  or  pauper,  is  an 
individual  equally  important  with  every  other  as  a  factor  in 
determining  the  result  we  are  seeking. 

In  the  subject  of  averaging  exchange-values  the  difficulty 
which  confronts  us  is  apparent.  The  individuals  Avith  which 
we  are  dealing  in  every  class  of  commodities  whose  prices  are 
reported,  are  not  directly  given  ;  for,  as  already  remarked,  it  is 
purely  accidental  what  the  actual  individuals  are  which  have 
prices  recorded  against  them.  The  price  of  wheat  is  now  in 
America  generally  reported  in  bushels.  In  England  it  used  to 
be  reported  in  quarters.  If  these  two  systems  be  applied  to 
the  same  mass  of  wheat,  the  former  would  report  the  price  of 
eight  times  as  many  individual  portions  of  wheat  as  the  latter, 
the  portions  referred  to  in  the  latter  being  eight  times  as  large 
as  those  in  the  former.  It  is  obvious  that  the  number  of  times 
the  price  of  a  bushel  of  wheat  is  repeated  has  no  more  right  to 
be  the  "weight"  of  the  wheat-price  in  our  calculations  than 
has  the  number  of  times  the  price  of  a  quarter  of  wheat  is  re- 
peated, or  reversely,  or  than  the  number  of  any  other  accident- 
ally chosen  mass-unit  of  wheat.      And  so  with  the   i)ric'fs,  and 


90       AKRAXGEMP^XT    OF    PARTICULAR    EXCHAXGE- VALUES 

the  numbers  of  individuals  wliioli  hear  them,  in  all  other  fungi- 
ble articles.  It  is,  therefore,  a  problem  what  are  the  individuals 
we  are  dealing  with.  This  question  must  be  answered  before 
we  can  find  the  sizes  of  the  classes,  which  sizes  depend  upon 
the  numbers  of  individuals  there  are  in  each  class. 

§  3.  Here  a  certain  position  above  reviewed  seems  to  offer  aid. 
This  is  the  position  that  we  ought  always  to  take  the  })rices  of 
the  same  mass-unit,  preferably  one  of  ^^'eight,  applied  to  all  com- 
modities, and  to  judge  the  sizes  of  the  classes  by  the  numbers 
of  this  mass-unit  they  contain.  According  to  this  doctrine,  the 
individual  in  all  classes  is  the  same  mass-quantity,  or  the  same 
s})ace-(piantity,  and  the  relative  sizes  of  the  classes  are  according 
to  their  relative  total  weights  or  volumes.  The  former  idea 
being  ado])ted,  the  "  weights  "  of  the  classes  in  our  calculations 
are  to  be  literally  according  to  the  weights  of  the  classes.  This 
is,  evidently,  a  very  convenient  doctrine. 

Against  it  some  objections  have  already  been  urged.  But  the 
principal  one  has  not  yet  been  noticed.  This  is  that  weighting 
so  determined  would  be  according  to  a  different  attribute  in  the 
things  from  the  one  we  are  measuring.  It  Avould  be  like  weight- 
ing the  calculation  of  the  average  tallness  of  the  three  sous, 
above  cited,  by  the  weight  or  girth  of  each  of  the  boys.  It  is 
plain  that  the  relative  weight  or  volume  of  two  classes  of  com- 
modities does  not  indicate  the  relative  importance  of  these  two 
classes.  We  have,  however,  agreed  not  to  judge  the  sizes  of 
classes  by  their  importance.  But  it  is  equally  plain  that  the 
relative  weight,  or  volume,  of  two  classes  does  not  indicate  their 
relative  sizes  from  the  point  of  view  in  which  we  have  the 
classes  under  consideration.  Their  relative  weights  indicate 
their  sizes  judged  by  weight.  Their  relative  volumes  indicate 
their  sizes  judged  by  volume.  What  indicates  their  sizes  judged 
as  things  possessing  exchange-value  ?  Evidently  only  the  rela- 
tive amounts  of  the  exchange- values  of  their  whole  quantities.^ 

1  Of  course  in  the  above  examples  we  would  not  weight  the  figures  according 
to  the  tallness  of  the  l)oys,  or  according  to  the  total  tallness  of  the  people  in  the 
different  districts.  This  is  because  in  those  examples  the  individuals  are  given, 
being  discrete  wholes.  The  proper  comparison  is  with  averaging  in  the  case  of 
things  with  uniform  substance  that  are  found  in  different  sizes.     In  averaging, 


WEIGHTING    EXI'LAIXHI)  91 

To  ])n)V('  tliis,  we  must  rcvicMV  the  nature  of  the  suhjcct  for 
wliicli  we  are  seeking-  the  weiglitinu:. 

§  4.  AVliat  we  iire  seeking  ultimately  is  the  combined  result 
on  tlie  general  exchange-value  of  something  when  we  know  the 
\-ariations  in  its  |)arti('ular  exchange-values.  "We  have  seen 
that  when  a  particular  exchange-value  of  the  thing  rises  to  a 
certain  height,  this  i-ise  may  be  compensated  by  another  of  its  par- 
ticular exchange-values  falling  to  a  certain  extent,  although  we 
have  not  yet  found  the  proportion  between  these  balancing  vari- 
ations. But  we  have  also  seen  that  if  two  of  its  other  particular 
exchange-values  fall  instead  of  only  one,  these  would  each  have 
to  fall  to  a  less  extent  to  counterbalance  the  one  rise  (in  Pro])o- 
sition  XXX.).  We  Avant,  then,  to  know  what  constitutes  one 
particular  exchange-value,  and  what  constitutes  two  particular 
exchange-values — that  is,  two  particular  exchange-values  that 
are  twice  as  large  (or  wide,  so  to  speak)  as  the  one.  Thus  far 
we  have  merely  treated  of  a  particular  exchange- value  as  one  if 
it  be  merely  an  exchange-value  viewed  in  relation  to  one  other 
exchange-value,  without  regard  to  the  size  of  either  of  the  t^o 
classes.  This  we  now  find  to  be  not  quite  adequate.  AVe  find 
the  need  of  another  distinction.  Let  us  keep  the  term  "partic- 
ular-value "  in  its  previous  usage,  and  introduce  for  what  Ave 
now  want  the  term  '^individual  exchange- value."  A  particular 
exchange-value  of  a  thing  is  its  exchange-value  in  one  other  class 
of  things ;  an  individual  exchange-value  is  its  exchange-value 
in  one  other  thing.  An  individual  exchange-value  relates  to  a 
particular  exchange-value  very  much  as  a  particular  exchange- 
value  relates  to  general  exchange-value  ;  for  as  many  particular 
exchange-values  compose  one  general  (or  generic)  exchange- 
value,  so  many  individual  exchange-values  compose  one  partic- 
ular (or  specific)  exchange-value.  Our  <|uestion  then  is,  What 
constitutes  one  individual  exchange-value,  whose  variation  is  to 
be  counterbalanced  by  an  opposite  variation  in  some  other  in- 
dividual exchange-value,  and  by  a  lesser  variation  in  other  two 
individual  exchange-values  ? 

for  instance,  the  yields  of  different  fields  (in  w  liieli  the  superficies  is  the  element 
of  importance)  in  order  to  find  the  average  yield  of  a  country,  we  should  count 
every  field,  not  as  one  individual,  but  according  to  the  relative  extent  of  its 
superficies. 


92       ARRANGEMENT    OF    PARTICUEAR    EXCHANGE- VALUES 

Now  when  one  thing  lias  a  certain  exchange-vahie  in  another 
thing,  it  is  eqnal  in  exchange-value  to  the  quantity  of  that  other 
thing  by  which  its  exchange-value  in  it  is  measured.  Hence  if 
the  thing  lA  is  equivalent  to  6B  and  to  cC  (these  capital  letters 
referring  to  mass-units,  and  the  small  letters  to  numbers),  we  see 
that  when  we  speak  of  A's  rise  in  exchange- value  in  B  being 
com])ensated  by  its  fall  in  exchange-value  in  C,  we  have  in  mind 
its  rise  from  equivalence  to  6  B  and  its  fall  from  equivalence  to 
cC,  wherefore  these  are  the  two  individual  exchange- values  we 
are  comparing  with  each  other  at  that  period.  Therefore  also  the 
two  individual  things  in  the  classes  B  and  C  with  which  we  are 
comparing  lA  are  6B  and  cC — its  equivalents  at  the  period  in 
question.  Similarly  if  1 A  be  equi^•alent  to  c  D,  the  economic  in- 
dividual in  the  class  D  with  which  we  are  com})aring  lA  is  cD. 
Suppose  the  rise  of  1 A  to  equivalence  with  b'b  B  is  compensated 

by  the  fall  of  lA  to  equivalence  with    7,  rC  {b'  and  c'  being 

certain  figures  larger  than  unity).  Then  if  A  falls  in  D  also, 
the  balance  is  disturbed.  Therefore,  for  the  balance  to  be  main- 
tained, when  A  falls   in   C   and  in  D   equally,  it  must  fall  in 

each  less  than  bv    ,.      Now  sui)i)ose  the  individuals  c  C  and  cD 

are  exactly  alike,  so  that  we  })iit  them  in  the  same  class  and 
refer  to  them  by  the  same  term,  say  Z.  Then  when  A  falls  in 
Z,  it  falls  both  in  C  and  in  D  equally.  Therefore,  to  counter- 
balance its  rise  in  B,  it  must  fall  less  in  Z  than  it  would  have 
to  fall  in  C  alone  or  in  D  alone.^  It  is  obvious  that  the  class 
Z  is  larger  than,  and  just  twice  as  large  as,  the  class  B,  at  the 
period  in  question,  Ixscause  it  embraces  twice  as  many  individuals 
as  the  class  B,  since  it  eml)races  two,  namely  e  C  and  c  D,  while  B 
embraces  only  one,  b  B.      And  so  if  B  and  Z  are  variously  large 

-  Proposition  XXX.,  it  sliould  be  noticed,  may  apply  both  to  other  chisses 
and  to  other  individuals.  It  may  mean  that  when  the  compensatory  changes  are 
in  a  greater  numl)er  of  (eqnal)  classes,  they  must  be  smaller,  and  wlien  they  are 
in  a  greater  number  of  (equal)  individuals  (in  the  same  class),  they  must  be 
smaller.  The  reason  is  that  in  the  former  case  as  in  the  latter  tlie  compensation 
is  distributed  over  more  individual  things.  It  is  plainly  indifferent  whether  the 
individuals,  their  number  being  given,  be  all  in  one  class  or  in  many  classes. 
The  important  factor  is,  7iot  the  imml)er  of  classes,  but  the  number  of  individuals. 


WETOTiTrNr;   explaixed  93 

classes,  their  relative  sizes  will  be  determined  by  the  iniiulxTs  of 
6  B's  and  of  c  Z's  tliev  eoiitaiii — that  is,  the  numbers  of  e(|iiivalents 
to  lA.  Generalizino-  still  more,  we  see  that  it  is  the  number  of 
equivalents  to  lA  in  any  class,  which  constitutes  its  relative 
size,  these  equivalents  to  lA  being  the  economic  indivlduah  in 
them  corresponding  to  the  posited  economic  individual  lA. 
And,  of  course,  whatever  be  the  quantity  lA,  a  large  or  a  small 
mass-unit,  tlie  ])rop()rtions  of  the  numbers  of  e(]uivaleuts  to  lA 
in  the  other  classes,  and  consecpiently  of  their  sizes,  will  be  the 
same  relatively  to  one  another  and  to  the  class  A.  Hence  if 
instead  of  lA  we  use  IM — a  money-unit,  say  one  dollar, — then 
the  relative  sizes  of  all  the  classes  of  commodities  are  })ropor- 
tioned  to  the  "  dollar's  worths  "  they  contain,  which  are  indicated 
by  the  total  money-values  of  these  classes. 

All  this  reasoning  is  obviously  independent  of  any  considera- 
tion of  the  mass-quantities  of  6B  and  of  rC,  su})])osed  to  be 
equivalent  to  lA,  or  of  the  mass-units  in  which  they  are  ex- 
pressed. If  lA  is  one  pound  of  wheat,  and  this  happened  to 
be  equivalent  to  one  pound  of  butter  as  b  B  and  to  one  ]>ound  of 
wool  as  c  C,  this  would  in  no  wise  improve  the  reasoning,  which 
would  be  the  same  if  the  one  pound  of  wheat  happened  to  be 
equivalent  to  two  pounds  of  butter  and  to  seven  and  three- 
quarter  pounds  of  wool.  On  the  other  hand,  if  this  last  hap- 
pens to  be  the  case,  it  is  plain  that  one  pound  of  butter  and  one 
pound  of  wool  have  no  claim  to  be  considered  economic  indi- 
viduals compared  with  one  pound  of  wheat.  As  masses,  one 
pound  of  butter  and  one  pound  of  wool  are  individuals  compared 
with  one  pound  of  wheat,  because  equal  to  it  in  weight.  But 
we  are  not  dealing  with  masses.  We  are  dealing  with  exchange- 
values.  And  the  exchange- value  of  one  pound  of  Avheat  bemg 
taken  as  our  given  individual,  it  is  only  the  quantities  of  butter 
and  of  wool  whose  exchange  values  are  equal  to  this  exchange- 
value,  at  the  period  in  question,  that  are  individuals  compared 
with  it,  in  our  economic  point  of  view. 

Or  if  we  compare  classes  with  money,  the  state  of  things  is 
still  j)lainer.  Supj^ose  cop])er  is  worth  ten  cents  a  ])ound  and 
iron  one  cent  a  pound.      It  is  evident   that   when   we  compare  a 


94       ARRANGEMENT    OF    PARTICULAR    EXCHANGE-VALUES 

variation  of  the  exchaui2:c-valiu'  of  one  dollar  in  e<)])])c'r  with  a 
variation  of  the  exchange-value  of  one  dollar  in  iron,  we  are 
comparino-  its  variation  relatively  to  ten  pounds  of  copper  and 
to  a  hundred  pounds  of  iron.  And  if  durino-  a  certain  period 
the  masses  of  copper  and  of  iron  with  which  we  are  dealing  are 
ten  million  pounds  of  the  former  and  fifty  million  pounds  of  the 
latter,  the  relative  sizes  of  these  classes — the  relative  nmnbers 
of  individual  exchange- values  of  IM  in  each  of  these  classes, 
that  vary  when  its  price-quotation  varies — is,  at  the  period  in 
question,  not  as  ten  for  copper  to  fifty  for  iron  (or  1  to  5),  but 
as  one  hundred  for  copper  to  fifty  for  iron  (or  2  to  1). 

It  will  be  noticed  that  this  explanation  does  away  with  de- 
pendence upon  the  idea  of  importance.  The  sizes  of  the  classes 
are  not  measured  by  their  importance  to  the  consumers.  They 
are  measured  by  the  numbers  of  equivalents  they  contain,  these 
equivalents  being  the  economic  individuals,  and  the  sizes  of  the 
classes  naturally  being  according  to  the  numbers  of  individuals 
they  contain.  It  is  true  that  the  importance  of  the  classes  is 
measured  in  exactly  the  same  way,  so  that  the  relative  sizes  of 
the  classes  go  hand  in  hand  with  their  relative  importance — and 
we  may  continue  to  speak  of  their  sizes  being  according  to  their 
importance.  But  it  is  not  directly  by  their  importance  that  the 
measurement  of  their  sizes  is  made.  It  is  made  by  the  measure- 
ment of  the  above  equivalent  individuals  they  contain.  This 
measurement  is  just  as  objective  as  the  measurement  by  the 
number  of  equiponderant  individuals  they  contain. 

§  5.  It  shonld  be  noticed,  further,  that  the  size  of  the  class 
whose  exchange-value  is  being  measured  is  of  no  consequence, 
and  therefore  need  not  be  considered,  except  only  in  the  one 
case  when  we  are  seeking  to  measure  a  variation  of  its  exchange- 
value  in  all  things.  For  if  we  are  seeking  to  measure  its  vari- 
ation in  exchange-value  in  all  oilier  things,  or  if  we  find  that  it 
has  remained  constant  in  exchange-value  in  all  other  things 
(and  consequently  in  all  things),  Ave  have  to  consider  only  the 
relative  sizes  of  the  oilier  classes,  the  size  of  the  class  in  ques- 
tion having  no  influence  in  the  result.  Thus  in  measuring  the 
variations  in  the  exchange-value  of  money,  we  are  dispensed 


HOME    DETAILS    IN    WEIGHTING  95 

from  enquiring  about  tlie  size  of  the  class  money  itself,  so  long 
as  we  confine  ourselves  to  measuring  its  variations  in  general 
exchange-value  in  all  other  things.  To  this  we  shall  confine 
ourselves  for  the  greater  part  of  our  researches,  and  shall  treat 
of  the  more  difficult  problem  of  measuring  variations  in  the 
general  exchange-value  of  money  in  all  things  only  at  the  end 
of  our  work,  where,  too,  we  shall  consider  M'hether  such  a 
measurement  is  needed  or  not. 

lY. 

§  1.  The  number  of  individuals  or  equivalents  m  each  class 
during  a  given  period  is,  of  course,  not  to  be  measured  by  the 
number  of  them  in  existence  at  any  one  moment.  For  articles 
are  variously  durable,  and  the  stocks  on  hand  at  any  moment  do 
not  represent  the  relative  quantities  in  which  they  are  ordinarily 
used.  These  quantities  can  only  be  ascertained  by  taking  all 
the  quantities  brought  into  trade  during  a  period  sufficiently  long 
to  cover  all  the  ups  and  do^vns  of  the  stocks — at  least  a  year  as 
being  the  shortest  natural  cycle.  If  land  be  one  of  the  things 
taken  into  account,  not  its  total  money-value,  n(jr  even  the  total 
money-value  sold  during  a  year,  but  its  annual  rental  is  the  item 
to  be  compared  with  the  total  money-values  of  the  commodity 
classes.^ 

§  2.  It  is  a  question  whether  the  total  money- values  of  com- 
modity classes  to  be  considered  are  those  of  the  quantities  an- 
nually produced  or  those  of  the  quantities  annually  consumed ; 
for  in  every  country  importation  and  exportation  disturb  the 
equality  betAveen  production  and  consumption  in  the  case  of 
many  articles.  Perhaps  it  would  be  best  to  count  both — that  is, 
in  any  one  case  either  the  total  product  or  the  total  consumption 
according  as  the  one  or  the  other  is  the  greater.  Thus  in  the 
United  States  we  should  count  all  the  wheat  produced  and  all 
the  tin  consumed,  and  in  England  all  the  wheat  consumed  and 
all  the  tin  produced.  There  is  overlapping  here  ;  but  this  is 
right,  because  the  English  and  the  American  economic  worlds 

^  For  a  similar  reason  it  is  only  the  variations  of  rent,  and  not  the  variations 
of  the  price  of  land,  that  are  to  be  counted. 


96       ARKAXGE>[ENT    OF    PARTICULAR    EXCHANGE- VAI.TES 

do  ovcrhi]).  This  double  countintr,  wliicli  takes  j)laee  in  the  two 
measurements,  could  be  avoided  by  making  the  measuremout  of 
the  exchange-value  of  money  in  the  two  countries  together,  if 
that  be  ])ossible  ;  but  then  there  would  be  similar  overlapping 
witli  other  countries,  which  could  be  avoided  agam  only  by  mak- 
uig  the  measurement  for  the  whole  world  at  large — for  then  we 
should  have  to  count  only  the  total  quantities  produced  or  con- 
sumed, witliout  counting  exports  and  imports  twice.^  Of  course, 
the  total  money-values  must  not  be  confined  to  the  total  money- 
values  sold.  A  bushel  of  wheat  consumed  by  a  farmer  is  as 
valuable  as  a  bushel  of  wheat  which  he  sells. 

§  3.  Articles  that  pass  through  various  stages  and  appear  in 
each  under  a  special  name — as  hemp,  rope  ;  wheat,  flour,  bread  ; 
wool,  yarn,  cloth,  clothing, — should  not  be  counted  as  a  distinct 
quantity  in  each  stage.  If  that  were  done,  as  some  articles  ap- 
pear in  more  forms  than  others,  the  sizes  of  these  would  be 
unduly  magnified.  One  method  of  treating  these  thmgs  would 
be  to  group  together  all  tlic  various  stages,  and  to  assign  to  this 
group  a  size  according  to  the  total  value  of  its  highest  forms  (or 
the  highest  counted).  Another  would  be  to  confine  attention  to 
two  stages  only.  But  in  the  second  stage  there  might  be  several 
branches,  according  to  the  kinds  of  things  the  raw  material  is 
made  into. 

§  4.  To  find  with  accuracy  the  total  money-value  and  the 
average  price  of  any  commodity  class  during  a  given  period  is  a 
statistical  task  of  considerable  difficulty  ;  but  the  theory  of  the 
operation  is  simple.  ( )f  all  the  prices  reported  of  the  same  kind 
of  article  the  average  to  be  drawn  is  the  arithmetic  ;^  and  the 
prices  should  be  Aveighted  according  to  the  relative  mass-quan- 
tities tliat  were  sold  at  them.  For  example,  if  there  is  sold 
twice  as  much  cloth  in  September  as  in   July,  tlie  |)rice  in   Sep- 

-Some  statisticians  have  counted  only  tliese  overlapi)ings,  and  liave  weighted 
the  classes  according  to  their  relative  siy.es  in  the  country's  exports  and  imports. 
If  this  weighting  be  employed  merely  in  an  attempt  to  measure  the  volume  of 
foreign  trade,  as  done  by  Giffen  and  De  Foville,  less  objection  can  be  found. 
But  if  it  be  employed  in  a  measurement  of  the  general  exchange-value  of  money, 
it  is  a  very  defective  kind  of  weighting.  Its  defect  has  been  noticed  by  Nasse,  B. 
104,  p.  332. 

3  .Jevons,  B.  15,  pp.  41,  43  ;  Walrus,  B.  (il,  p.  (J. 


THE    QUESTION    OF    PERIODS  97 

tembcr  should  be  jrivt'ii  twice  as  niiicli  weig'lit  as  the  price  in 
July.' 

V. 

§  1.  As  yet  Ave  have  examined  the  sizes  of  the  classes  only  at 
a  single  j^eriod.  But  in  any  enquiry  into  the  variation  of  the 
general  exchange-value  of  anything  we  are  concerned  with  at 
least  two  periods.  Between  the  two  periods  not  only  tlie  par- 
ticular exchange-values  may  have  varied,  but  also  the  sizes  of 
the  classes.  And  the  sizes  of  the  classes  vary  partly  because  of 
the  very  variations  in  the  particular  exchange- values  whose  com- 
mon variation  we  are  attempting  to  measure.  The  difficulty  in 
the  subject  of  weighting,  in  the  case  of  averaging  exchange- 
values,  is  only  commencing. 

The  older  writers,  from  Arthur  Young  on,  who  touched  upon 
weighting,  recommended  merely  a  rough  weighting  according  to 
the  importance  of  the  classes  as  vaguely  measured,  by  their 
relative  total  money-values,  or  by  masses,  over  a  number  of 
years,  although  a  few,  using  custom-house  returns,  actually  did 
use  varying  weighting,  without  understanding  its  nature.^  The 
need  of  paying  attention  more  minutely  to  the  weighting  of  each 
period  compared  was  first  pointed  out  about  thirty  years  ago  by 
the  German  philosopher  and  mathematician,  Drobisch.  Drobisch 
was  the  originator  of  the  idea  that  we  should  weight  the  classes 
according  to  their  actual  physical  weights,  and  he  confined  his 

*  E.  Segnitz  gave  this  formula  for  finding  one  average  price, 

jy_pq+p'q'  +  p"q"+--- 
q  +  q'+q"+--.      ' 

in  which  pp'p"  represent  the  prices  at  which  respectively  qq'q"  quantities  of  the 
article  sold  during  the  period  in  question,  Ueber  die Berechnung  der  sogenannten 
Mittel  soicie  deren  Anwendung  in  der  Statistik  und  anderen  Erfahrungsivissen- 
schaften,  .Tahrbiicher  fiir  Nationaloekonomie  und  Statistik,  1870,  Band  XIV.,  p. 
184.  E.  Heitz,  though  not  approving  of  it  himself,  says  this  method  is  employed 
in  the  statistical  bureaus  of  several  German  states  (Wiirtemberg,  Bavaria,  Gotha, 
and  partly  in  Prussia  and  Hannover),  Ueber  die  3Iethoden  bei  Erhebung  der 
Preisen,  in  the  same  .lahrbiicher,  1876,  Band  XXVII. ,  pp.  347-351.  This  use  of 
uneven  weighting  has  been  recommended  also  by  Marshall,  B.  93,  p.  374,  and  the 
British  Association  Committee,  First  Report,  B.  99,  pp.  250-251. — Weighting  ac- 
cording to  the  length  of  time  each  price  lasts  was  recommended  by  Beaujon,  B. 
95,  p.  116. 

1  See  Appendix  C  IV  §  2  (2). 

7 


98       ARRANGEMENT    OF    PARTICULAR    EXCHANGE- VALUES 

attention  to  differences  at  the  two  periods  between  the  mass- 
<|iiantities  of  the  classes.  Considering  the  single  case  of  a  com- 
parison between  two  contignons  periods,  he  raised  the  following 
qnestions : — Shall  we  employ  in  our  calculations  the  mass- 
quantities  of  both  the  periods  separately?  or  only  the  mass- 
quantities  of  one  of  the  periods  ?  and  in  the  latter  case  shall  it 
be  those  of  the  first  period,  or  those  of  the  second  ?  or,  again, 
shall  we  combine  these  two  systems,  and  employ  a  single  one 
that  is  a  mean  between  them  ?  Between  these  questions  Dro- 
bisch  decided  in  favor  of  the  first  course  suggested.  In  doing 
so  he  originated  what  may  be  called  a  method  of  double  weight- 
ing. This  consists  in  drawing  an  average  of  the  prices  at  each 
period  separately,  and  at  each  period  on  the  mass-quantities  of 
that  period  ;  and  in  then  comparing  these  averages.  At  each 
period  conqjared  the  mass-quantities  may  be  measured  in  dif- 
ferent ways.  The  way  Drobisch  chose  was,  as  we  have  seen,  to 
measure  them  by  the  number  in  every  class  of  a  common  mass- 
unit.  Upon  the  method  of  double  weighting  he  decided  on  the 
gromid  that  the  mass-quantities  of  neither  of  the  two  periods  is 
preferable  the  one  to  the  otlier,-  and  that  there  is  no  better  reason 
for  the  mean  between  the  two,^  apparently  considering  that  the 
course  he  adopted  was  the  only  one  not  exposed  to  objections. 
Drobisch  has  found  but  few  followers  in  the  use  of  double  weight- 
ing, though  some  involuntary  ones  are  among  them.*  His  posi- 
tion was  immediately  attacked  by  Professor  Laspeyres,  for  a 
reason  which  will  be  noticed  later — a  reason,  however,  which  is 
valid,  not  against  his  use  of  double  weighting,  but  against  his 
use  of  a  common  weight-unit ;  and  a  little  later  by  Paasche,  on 
the  ground  that  we  must  use  the  mass-quantities  of  only  one 
period  so  as  to  get  the  variation  in  the  sums  of  money  needed  to 
purchase  the  same  mass-quantities  at  both  the  periods.^  And  a 
similar  reason  has  been  given  more  recently  by  Professor  Falk- 
ner,  namely  that  "  Ave  must  compare  like  with  like,"  *" — although, 

2  B.  29,  p.  39. 
3B.  31,  pp.  423-425. 
*  See  Appendix  C  V. 
^B.  33,  pp.  171-173. 
«B.  112,  p.  63. 


THE    QUESTION    OF    PEIU(n).S  99 

if  the  like  does  not  exist  at  both  the  periods,  the  comparison 
may  appear  somewhat  forced.  Laspeyres  recommended  using  the 
mass-quantities  of  the  first  period,  Paasche  those  of  the  second/ 
The  mean  between  these  two  positions  may  l)e  either  by  making 
two  eak'ulations,  the  one  on  the  mass-quantities  of  the  first 
period,  the  otlier  on  those  of  the  second,  and  then  talcing  the 
(arithmetic)  mean  between  the  two  results  ;  or  by  making  a 
single  calculation  with  mass-quantities  that  are  the  (arithmetic) 
mean  between  those  of  the  two  periods.  The  last  has  sometimes 
been  recommended,  though  only  half-heartedly.'^  Thus  every 
obvious  position  seems  to  have  been  occupied.  Amidst  all  this 
diversity  of  opinion  Professor  Sidgwick  has  asserted  that  as  we 
cannot  decide  in  favor  of  any  of  these  methods,  the  use  of  the 
mass-quantities  of  the  one  period  being  as  good  as  of  the  other, 
and  the  mean  between  them  having  no  "  practical  significance," 
and  as  the  answers  obtained  by  each  method  may  differ,  we  can 
therefore  make  no  one  authoritative  measurement  of  the  varia- 
tion of  money's  general  exchange-value.-'  And  the  same  criti- 
cism has  recently  been  reaffirmed  by  Dr.  Wicksell,  A\'ho  says 
that  the  problem  is  insolvable  unless  it  happens  that  the  same 
result  is  yielded  by  using  the  mass-quantities  of  each  period 
separately,  the  use  of  the  mean  between  the  mtvss-quantities  at 
the  two  periods  having  nothing  but  a  "  conventional  meaning."  '" 
The  truth  is,  however,  that  these  are  not  the  only  positions 
that  present  themselves,  and  the  subject  has  hitherto  been  treated 
most  inadequately.  Like  Drobisch,  the  writers  who  have  no- 
ticed this  question  have  considered  only  the  difference  at  the  two 
periods  between  the  mass-quantities.  But  it  is  strange  that  per- 
sons who  have  not  foUow^ed  Drobisch  in  adopting  mass-quanti- 
ties as  the  test  of  weighting,  and  Av^ho  have  even  asserted,  many 
of  them,  that  weighting  should  be  according  to  the  total  exchange- 
values  of  the  classes,  when  they  come  to  discuss  the  question 
concerning  divergent  weights  at  the  two  periods,  should  confine 

"  Pasche  has  been  followed  by  v.  d.  Borght  and  Conrad,  and  without  depend- 
ence by  Mulhall,  by  Sauerbeck  at  times,  and,  following  the  latter,  by  Powers. 
See  Appendix  C  IV  ?  2  (2). 

8  See  Appendix  C  IV  ^  2  (3). 

9  B.  56,  pp.  67-68. 
"B.  139,  pp.  8-9. 


100    AKRAXCiEMEXT    OF    rAETICULAR    EXCHANGE- VALUES 

tlicii"  attention  t(»  ditferonccs  that  may  exist  between  tlie  mass- 
quantities  at  the  two  periods."  Changes  in  the  mass-quantities, 
and  changes  in  the  weights,  of  the  classes  are  two  distinct  things, 
and  the  former  are  of  importance  only  as  they  affect  the  latter. 
Since  no  (ine  has  so  much  as  noticed  the  question  in  its  j)roper 
aspect,  we  are  left  to  our  own  devices,  and  must  investigate  the 
subject  06  ovo. 

§  2.  There  are  three  variables  :  (1)  the  exchange- values,  or 
prices,  of  given  mass-nnits  of  many  classes  of  commodities  ;  (2) 
the  total  mass-quantities  of  these  classes ;  (3)  their  total  ex- 
change-values, or  money-values.  The  first  are  the  ones  w^e  are 
trvins:  to  averaoe.  The  second  and  third  are  factors  in  the 
weighting  of  the  classes  in  our  averaging  of  their  exchange-values 
or  prices,  or  of  the  variations  of  these.  Yet  the  third  itself  is 
dependent  upon  the  first  two,  being  their  product.  As  we  are 
now  examining  only  the  factors  that  enter  into  the  weighting  of 
the  classes,  we  are  not  concerned  with  the  first  variations  except 
so  far  as  they  affect  the  third. 

With  the  two  variables  that  enter  into  weighting  we  have, 
then,  two  distinct  elemental  cases,  and  a  third  composed  of  these 
two.  The  first,  and  simplest,  is  when  the  total  sums  spent  on 
every  class  remain  constant  over  both  the  periods,  so  that  the 
relative  sizes  of  the  classes  remain  the  same  at  both  the  ])eriods, 
the  variations  of  the  ])rices  being  counteracted  by  inverse  vari- 
ations of  the  mass-quantities.  The  second  is  when  the  mass- 
quantities  in  every  class  remain  constant  over  both  the  periods, 
so  that  the  sizes  of  the  classes  vary  from  period  to  period  directly 
as  the  prices.  The  third  is  when  both  the  mass-quantities  and 
the  total  sinns  s])ent  on  them  vary  from  period  to  ])eriod. 

§  3.  I.  In  this  sim])le  case  the  relative  inq)ortance,  or  the 
relative  sizes,  of  the  classes  remain  constant  at  each  of  the 
periods,  whether  the  exchange-value  of  money  be  the  same  at 
both  the  periods  or  not.  It  would  therefore  seem  natural  to 
treat  the  classes  as  having,  in  our  averaging  of  their  price-vari- 
ations, the  relative  sizes  that  they  have  at  each  of  the  periods. 

^*  Giffen  and  Palgrave  are  exceptions.  The  former  used  weighting  according 
to  the  money-values  of  a  single  period,  the  latter  according  to  those  of  every  later 
period.     But  each  has  done  so  without  discussing  the  question. 


THE    QUESTION    OF    PEEIODS  101 

Thus,  for  exani])l(',  if"  :it  the  first  period  the  total  moii('v-\  aliic 
of  the  class  A  is  twice  as  great  as  the  total  money-value  of  the 
class  B,  aucl  if  at  the  second  period  each  of  these  classes  has  the 
same  total  money-value  as  before, — f)r  even  if  they  have  both 
varied  in  their  total  money-values,  but  in  the  same  ])ro|)()rtion, 
so  that  the  class  A  still  has  twice  the  money-value  (and  consc- 
qnently  twice  the  exchange-value,  as  may  be  deduced  from 
Proposition  VII.)  as  the  class  B, — we  should  look  upon  the  class 
A  as  twice  as  large,  or  as  containing  twice  as  many  economic 
individuals,  as  the  class  B. 

Yet  each  of  the  economic  individuals  in  these  classes  would 
consist  at  each  period  of  a  different  mass-quantity  of  the  article. 
When  this  is  perceived,  we  may  hesitate  to  adopt  this  position. 
We  may  best  discuss  this  question,  however,  when  we  are  ex- 
amining the  more  general  and  more  complex  cases  where  both 
the  total  exchange-values  and  the  total  mass-quantities  are  vari- 
able. 

§  4.  II.  The  next  simplest  case  is  when  it  hap})cns  that  the 
mass-quantities  of  the  various  classes  remain  constant  over  both 
the  periods,  and  their  prices  varying,  their  total  money-values 
are  different  at  each  period.  It  is  plain  that  this  supjwsitional 
case  escapes  all  the  questions  that  have  hitherto  been  raised  on 
this  subject,  above  reviewed  ;  and  yet  the  qnestion  of  weighting 
is  still  before  us.  For  the  sizes  of  the  classes  are  different  at 
each  of  the  periods  directly  according  to  the  variations  in  their 
relative  prices.  Hence  we  have  alternative  positions  like  those 
already  noticed,  except  that  here  there  is  no  opportunity  for 
double  weighting.  Shall  we  use  the  weighting  of  only  one  of 
the  periods  ?  and  then  shall  it  be  that  of  the  first  or  that  of  the 
second  period?  or  shall  we  use  a  mean  between  the  two  sys- 
tems? And  there  is  still  another  possible  position.  Shall  we 
use  only  the  smaller  total  money-value  that  occurs  at  either 
period  as  the  weight  of  the  class  for  l)oth  the  periods  together, 
eliminating  what  is  in  excess  at  one  of  the  periods? 

Plainly  there  is  no  reason  why  in  general  we  should  choose 
the  weighting  of  one  of  the  periods  in  preference  to  that  of  the 
other.      And  there  is  good   reason  for  rejecting  each  of  them — 


102    ARRANGEMENT    OF    PARTICULAR    EXCHANGE-VALUES 

in  crencral,  unless  later  discussion  of  the  avera<>es  shows  that  the 
one  fits  one  of  the  averages  and  the  other  another.  For  if  one 
class  happens  to  be  more  important,  or  larger,  than  another  at 
the  first  period,  and  at  the  second,  through  a  fall  in  the  price  of 
the  former  and  a  rise  in  the  price  of  the  latter,  is  less  impor- 
tant, or  smaller,  it  Avould  be  absurd  to  take  either  of  these  con- 
ditions as  alone  representative  of  the  proper  relationship  be- 
tNveen  these  classes.  Or  if  two  classes  are  equally  large  in  total 
exchange-valne  at  the  first  period  and  the  one  becomes  larger 
and  the  other  smaller  at  the  second  period,  and  if  two  other 
classes,  at  the  first  period  the  one  larger  and  the  other  smaller, 
become  equal  in  size  at  the  second  period,  it  is  plain  that  we 
have  no  better  right  to  treat  the  classes  in  the  one  of  these  pairs 
as  equal  in  size  than  to  treat  the  classes  in  the  other  as  such ; 
but  it  is  impossible  to  do  so  with  both,  if  we  are  to  use  the 
weighting  of  only  one  period — all  which  shows  that  the  classes 
in  neither  of  these  pairs  are  to  be  treated  as  equal,  yet  not  so 
unequal  as  in  each  they  are  at  one  of  the  periods. ^^ 

§  0.  Therefore  Ave  must  use  either  a  mean  between  the  weight- 
ing of  the  two  periods,  or  the  weights  partly  of  the  one  and 
})artly  of  the  other,  picking  out  the  smaller  weight  of  a  class  at 
either  period.  The  reason  for  entertaining  such  a  position  as 
the  last  is  obtained  from  the  preceding  Chapter.  We  have  seen 
a  desideratum  to  be  that  we  should  have  the  same  or  a  similar 
whole  at  l)otli  ])eriods,  in  which  to  compare  the  relationshij)  be- 
tween a  part  and  the  whole  of  which  it  is  a  part.  An  economic 
individual  being  an  exchange- value  quantum  rather  than  a  mass 
quantum,  it  may  be  maintained  that  we  ought  to  eliminate  all 
those  individuals  which  exist  at  one  ])eriod  alone,  which  can  be 
done  by  taking  the  total  exchange-value  that  is  the  smaller  at 
either  period.  The  mass-(|uantities  being  sup])osed  in  all  cases 
constant  over  both  periods,  if  the  class  A  rises  in  exchange- 
s-It may  be  objected  that  it  is  only  the  individuals  at  the  first  period  that 
vary,  the  individuals  at  the  second  period  being  merely  a  result  of  the  variation  of 
tlie  former.  But  really  the  variation  is  from  the  individuals  at  the  first  period  to 
tlie  individuals  at  the  second  period  ;  and  tlie  number  of  individuals  that  have 
varied  is  not  the  number  existing  at  the  first  period  any  more  than  it  is  the 
number  existing  at  the  second. 


THE    QUI^STIOX    OF    PERIODS  103 

value  Ix'twcHMi  the  first  and  the  second  ])eriod,  its  total  exeliano-e- 
value  increases,  and  only  that  M'hicli  it  liad  at  the  first })eriod 
should  be  considered;  and  if  the  class  B  falls  in  exchange- 
value,  it  contained  at  the  first  period  more  economic  individuals 
than  at  the  second,  and  so  only  those  existing  at  the  second 
period  should  be  considered. 

A  practical  objection  to  this  position  is  that  the  economic  indi- 
viduals at  each  period  should  be  the  same,  that  is,  have  the  same 
exchange-value,  or  be  ecpiivalents  to  a  money-unit  having  the 
same  exchange-value  at  both  the  periods ;  but  whether  the 
money -imit  has  the  same  exchange- value  at  the  second  period 
as  at  the  first,  is  the  question  to  be  determined.  Merely  to 
compare  the  total  money-value  of  the  classes  at  the  two  periods 
is  not  to  compare  their  total  exchange-values  ;  yet  that  is  all  we 
can  do  till  we  have  found  the  constancy  or  variation  of  the  ex- 
change-value of  money.  It  is  plain  that  if  the  exchange-value 
of  money  be  falling,  i.  e.,  if  prices  in  general  be  rismg,  the 
smaller  total  money-values  might  all,  or  most  of  them,  occur  at 
the  first  period,  and  then  in  selecting  them  for  our  weights,  we 
should  be  taking  merely,  or  mostly,  the  weights  of  the  first 
period  ;  or  if  the  exchange-value  of  money  be  rising,  i.  c,  if 
prices  in  general  be  falling,  the  weights  might  be  mostly  those 
of  the  second  period.  Or  between  one  country  and  another, 
where,  the  mass-quantities  being  the  same,  the  levels  of  prices 
are  considerably  different,  we  might  be  really  taking  the  weights 
almost  altogether  from  those  of  one  of  the  countries  alone,  namely 
the  one  in  which  the  level  of  prices  is  the  lower.  It  is  perhaps 
possible  that  we  might  approximate  to  the  ultimate  result  by 
several  stages  of  approach.  Assuming  at  first  that  money  is 
stable,  we  should  take  the  total  money-value  as  representing  the 
total  exchange-values  of  the  (!lasses,  and,  choosing  the  smaller 
ones  at  either  period,  should  use  these  as  the  weights  of  the 
classes ;  then  if  the  result  showed  constancy  in  the  exchange- 
value  of  money,  we  should  rest  content ;  but  if  it  showed  a  vari- 
ation, we  should  take  this  as  a  means  of  correcting  the  estimate 
of  the  total  exchange-values  of  the  classes  at  the  second  period 
compared  with  what  they  Avere  at  the  first,  and  rei)eat  the  proc- 


104    AKHAXGEMEXT    OF    PARTICULAR    EXCHANGE- VALUES 

ess,  and  keep  on  doing  so  till  the  last  resnlt  tallies,  or  nearly 
tallies,  with  the  preceding.  But  there  is  no  certainty  in  this 
very  laborious  method. 

A  more  theoretical  objection,  however,  exists.  This  is  di- 
rected against  the  very  foundation  upon  which  the  position  rests. 
The  ])osition  involves  the  conception  that  we  are  measuring  the 
variation  in  purchasing  power  of  a  certain  total  sum  of  money 
(or  of  sums  of  money  having  the  same  general  exchange-vahu!) 
over  a  certain  class  of  commodities,  and  the  variation  in  the  pur- 
chasing power  of  another  certain  sum  of  money  over  another 
class,  and  so  on,  and  are  seeking  to  average  these  variations. 
But  this  does  not  correctly  represent  what  we  are  seeking.  It 
represents  a  method  of  constructing  a  similar  world  at  both 
periods  like  the  one  Avliich  in  the  preceding  Cha})ter  itself  was 
subordinated  to  another  method.  What  we  are  seeking  is  to 
average  the  variations  in  the  exchange-value  of  one  given  total 
sum  of  money  in  relation  to  the  several  classes  of  goods,  to 
which  several  variations  must  be  assigned  weights  proportional 
to  the  relative  sizes  of  the  classes.  Hence  the  relative  sizes  of 
the  classes  at  both  the  periods  must  be  considered.  And  now, 
if  for  instance  at  the  first  period  one  class  be  three  times  as  large 
as  another,  and  at  the  second  the  latter  be  twice  as  large  as  the 
former,  it  is  impossible  to  say  which  are  the  relative  sizes  that 
are  common  to  both  the  periods.  We  are  left  with  the  position 
requiring  a  mean  between  the  relative  sizes  at  both  periods. 
And  the  question  arises,  what  kind  of  a  mean  is  this  to  be? 

§  ().  At  first  sight  it  might  be  thought  sufficient  to  add  up  the 
weights  of  every  class  at  the  two  periods  and  to  divide  by  two. 
This  would  give  the  (arithmetic)  mean  size  of  every  class  over 
the  two  periods  together.  Hut  such  an  operation  is  manifestly 
wrong.  In  the  first  ])lace,  the  sizes  of  tiie  classes  at  eaeli  jieriod 
are  reckoned  in  the  money  of  the  ])criod,  and  if  it  happens  that 
the  exchange-value  of  money  has  fallen,  or  prices  in  general 
have  risen,  greater  influence  upon  the  result  would  be  given  to 
the  weighting  of  the  second  period  ;  or  if  ]irices  in  general  have 
fallen,  greater  influence  would  be  given  to  the  weighting  of  the 
first  period.     Or  in  a  comparison  between  two  countries  greater 


THE    QUESTION    OF    PERIODS  105 

influence  would  be  given  to  the  weighting  of  tlie  country  with 
the  higher  level  of  prices.  But  it  is  plain  that  the  one  j^eriod, 
or  the  one  country,  is  as  important,  in  our  comparison  between 
them,  as  the  other,  and  the  weiffhtinr/  in  the  averaffnu/  of  their 
weights  should  rea//i/  be  even.  To  employ  here  the  same  sort  of 
correction  by  approach  as  suggested  in  the  similar  case  preceding 
would  not  only  be  extremely  laborious,  but  ^vonld  be  useless  be- 
cause of  the  next  reason.  In  the  second  place,  even  if  the  ex- 
change-value of  money  happens  to  remain  stable,  or  to  have 
varied  so  slightly  as  not  to  be  appreciable  in  this  way,  as  may 
often  occur,  this  operation  can  be  justified  only  on  the  supposi- 
tion that  the  problem  is  that  of  putting  together  the  economic 
individuals  of  the  first  period  and  the  economic  individuals  of 
the  second,  and  of  then  seeing  how  the  combined  quantities  re- 
late to  one  another.  But  this  is  not  the  })roblem,  because,  l)V 
the  hypothesis,  the  two  periods  are  distinct,  and,  while  they  can 
be  compared,  they  cannot  be  united.  Our  only  data  are  prop- 
erly the  relations  that  the  class  A  is  so  many  times  more  or  less 
important,  or  large,  than  the  class  B  at  the  first  period,  aud  at 
the  second  so  many  other  times  more  or  less  important,  or  large. 

Nor  is  the  simple  arithmetic  mean  of  these  relatif)ns,  l)y  add- 
ing them  together  and  dividing  l)y  two,  the  proper  one  to 
draw.  The  proper  mean  in  cases  of  this  sort  is  the  geometric. 
Of  course,  as  above  remarked,  we  must  employ  even  weighting 
in  drawing  this  mean. 

The  reason  why  the  arithmetic  mean  is  improper  and  the 
geometric  mean  is  proper  is  a  purely  mathematical  one,  which 
will  be  examined  more  fully  in  a  later  Chapter.  In  general,  the 
operation  of  adding  the  separate  terms  in  ratios  for  the  purpose 
of  getting  the  average  ratio  is  imallowable,  although  it  may  be 
correct  in  sp(^cial  instances.  The  operations  to  be  performed 
with  ratios  are  princi[)ally  division  and  multiplication.  In  par- 
ticular, the  true  position  in  the  case  before  us  may  be  shown  as 
follows.  Let  the  class  A  at  the  first  period  be  r^  times  as  large 
as  the  class  B  and  s^  times  as  large  as  the  class  0.     Then  at  tliis 

period  the  class  B  is    '  times  as  large  as  the  class  C.     And  at 


1 


106    ARRAXGEMENT    OF    PARTICULAR    EXCHAXGE-YALUES 

tlie  second  period  let  the  classs  A  be  i\,  times  as  lariie  as  the  class 
B  aud  s.^  times  as  large  as  the  class  C.     Then  at  this  period 

the  class  B  is  ^  times  as  large  as  the  class  C.  Now  if  the  arith- 
metic mean  be  nsed  in  averaeing;  these  relations  in  order  to  ffet 
the  size  relationship  for  both  the  periods  together,  we  should  have 
to  say  that  A  is  |(r^  -|-  r^)  times  as  large  as  B  and  ^(.Sj  +  s^) 
times  as  large  as  C.  Then  if  we  draw  the  relationship  between 
B  and  C  from  these  relations,  we  should  have  to  say  that  B  is 

-^ ^  times  as  lar^e  as  C.     But  if  avc  draw  the  relationship 

n  +  r,  ^  i 

between  B  aud  C  in  the  same  way  from  the  first  separately 
given  relations  between  B  and  C,  we  should  have  to  say  that  B 

A  -  -r  ---  J  times  as  large  as  C.  But  this  expression  is  not 
the  same  as  the  preceding,  and  is  equal  to  it  only  in  special  cir- 
cumstances I  namelv  if  r,  =  r.„  or  if  —  =~  ).     Therefore  the 

employment  of  the  arithmetic  mean  involves  an  inconsistency, 
M'hich  shows  it  to  be  absurd.  The  geometric  mean  is  free 
from  this  inconsistency.  This  mean  being  employed,  the  size 
relationships  for  both  the  periods  together  are  to  be  expressed 
by  saying  that  A  is  ^y^r^^r^  times  as  large  as  B  and  V^Sj-Sj 
times  as  large  as  C,  wherefore,  using  these  relations,  we  must 

V^s  ■  s 
sav  that  B  is  — .  ^     ^-  tunes  as  large  as  C.     And  if  we  used  the 

V7\-r, 

first  separately  given  relations  between  B  and  C,  we  should  have 

to  say  that  B  is      f  ^-  .    -  times  as  large  as  C  ;  which  expression 

^'\      >'i 
ditfers  from  the  ])receding  only  in  form,  and  is  universally  c({ual 

to  it. 

With  mauy  classes,  this  operation  of  geometrically  averaging 

the  numbers  of  times  each  is  more  important  than  another  would 

be  interminable.      It  can  be  simj)lified.      All  we  have  to  do  is 

to  draw  the  f/eonietric  mean  of  the  iceights  of  cverij  cJasn  at  the  two 

psrlods,  and  to  take  the  results  as  the  weights  of  the  classes  for 

both  the  periods  tof/rthrr.     The  relative  weights  so  obtained  are 


THE    QUESTION    OF    PERIODS  107 

the  same  as  if  we  compared  every  class  with  every  otlicr  in  the 
way  described.  This  may  be  shown  as  follows.  The  total  mass- 
quantity  of  the  class  A,  which  is  supposed  to  be  the  same  at  both 
the  periods,  may  be  represented  by  .r,  and  the  total  mass-quantity 
of  the  class  B  by  //  (x  and  y  standing  for  certain  numbers  of  cer- 
tain mass-units  of  A  and  B,  what  these  mass-units  are  being  in- 
different). And  the  price  (of  the  mass-unit  before  used  for  the 
quantity)  of  A  may  be  represented  by  a^  at  the  first  period  and 
by  a^  at  the  second  ;  and  the  price  (of  the  mass-unit  before  used 
for  the  quantity)  of  B  may  be  represented  by  /9j  at  the  first 
period  and  by  j^.-^  at  the  second.  Then  the  weight  of  the  class  A 
is  .r«j  for  the  first  period,  and  for  the  second  xa.^ ;  and  the  weight 
of  the  class  B  is  _?//?j  for  the  first  period,  and  for  the  second  yi3.^. 
Let  .r«j  =  /-J  •  ?/;9j,  and  xa^  =  /■.,  •  yi^^-  Then  by  the  above  method, 
the  weight  of  the  class  A  for  both  the  periods  is  Vi\  ■  r^  times 
the  weight  of  the  class  B.  Now  the  geometric  mean  of  the 
weights  at  both  periods  of  the  class  B  is  1  ^^;5i  •  yf^l^  =  y^^i^it^z ' 
and  that  of  the  two  weights  of  the  class  A  is  V  )\yj3^^  ■  r^j^^  = 
3/l/'^/9,/5,  •  r^r.,.     The  relationship  between  these  is 


weight  of  A  for  both  periods     VV  ^^^^  ■  r^r.^         / 

weight  of  B  for  both  periods      yVR^_  ^  ' ' 

the  same  as  before.  AVhat  is  here  demonstrated  of  two  classes, 
may  be  demonstrated  in  the  same  way  of  three  classes,  of  four 
classes,  and  of  any  number  of  classes. 

It  is  plain  also  tliat  this  method  is  unaffected  by  the  possible 
variations  in  the  exchange-value  of  money,  or  the  general  level 
of  prices.  For,  statically  considered,  the  variation  of  the  ex- 
change-value of  money  affects  all  prices  alike.  Then,  at  the 
second  period,  if  the  prices  of  the  mass-unit  of  A  is  ta^  instead 
of  «,,  the  price  of  the  mass-unit  of  B  will  be  tfi,  instead  of  jd^^ 
and  so  on.  This  t  will  ai)pear  as  I'^t  in  the  weight  of  every 
class  for  both  the  periods  together;  and  so  will  not  alter 
the  relationship  between  these  weights,  disappearing  in  the 
comparisons  just  as  l'^/?,/?^  has  done.  Hence  even  weighting  is 
here  really  carried  out  in  our  averaging  of  the  weights  of  every 
class  at  both  periods,  or  in  two  ccjuntries. 


108    ARRANGEMENT    OF    PARTICULAR    EXCHANGE-VALUES 

§  7.  AVe  may  illustrate  this  ])()siti()ii  l)y  an  example,  Avliich 
■will  not  only  render  it  clearer,  but  will  also  disclose  certain  im- 
portant details.  Suppose  at  both  periods  we  have  seven  pounds 
of  A  and  seven  pounds  of  B  (or  any  greater  quantities  iii  the 
same  proportion),  and  that  at  the  first  period  each  of  these  seven 
pounds  is  valued  at  100  dollars,  but  that  at  the  second  period 
the  money-value  of  the  seven  pounds  of  A  is  196  dollars,  and 
the  money -value  of  the  SQ\'en  pounds  of  B  is  4  dollars.  (Such 
extreme  variations  are  purposely  chosen,  because  they  bring  out 
the  distinction  more  clearly.)  Then,  for  both  the  periods  to- 
gether the  class  A  is     I =  7  times  larger  than  the  class  B, 

— which  relationship  is  also  shown  liy  the  fact  that  1    19600  = 

140 

140  and  1/400  =20,  and  -     =  7.    Then  it  is  proper  to  say 

that  over  these  two  ])eriods  together  the  class  A  contains  seven 
economic  individuals,  and  the  class  B  one  economic  individual  ; 
which  means  that  the  economic  individual  in  A  consists  of  one 
pound  of  A  and  the  economic  individual  in  B  consists  of  seven 
pounds  of  B.  -Now  the  money-value,  or  price,  of  the  eco- 
nomic individual  in  A,  namely  one  pound  of  A,  is  $14.2857 
at  the  first  period,  and  the  second  $28.00,  while  the  money- 
value,  or  price,  of  the  economic  individual  in  B,  namely  seven 
pounds  of  B,  is  $100.00  at  the  first  ])eriod,  and  at  the  second 
$4.00.  Thus  the  individual  in  A  is  at  the  first  period  one 
seventh  as  large,  and  at  the  second  seven  times  as  large,  as  the 

individual  in  B  ;  for  14.2857  =  ^   ,  and  28  =  4x  7.     And  the 

geometric  mean  ])rice  of  these  individuals  at  both  the  periods 
is  the  same,  being  in  each  case  $20.00;  for  l/l4.2857x28 
=  l/T0(b^4  =  20. 

These  relations  are  universal,  as  may  be  demonstrated  in  the 
fi)llowing  simple  manner.  Using  the  same  symbols  as  above, 
the  total  money-values  of  the  class  A  are  xa^  at  the  first  period 
and  .ra.,  at  the  second,  and  those  of  the  class  B  i/j3^  and  ?//?2  at 


the  two  periods  respectively.      Therefore,  the  class  .V  i 


TIIK    t^UESTIOX    OF    PERIODS  109 

times  larg'cr  (or  smaller)  than  the  class  B,  or,  iu  other  >vor(ls,  if 
the  class  B  be  regarded  as  containing-  one  economic  individnal, 
being  taken,  so- to  speak,  as  the  unit-class,  the  class  B  contains 

*_     f   '-j  individuals.     Then  the  moncv- value,  or  price,  of  tlie  in- 


dividual  in  the  class  A  is    ^^-^  =     \^  .  y  Vj%B.,  at  the  first 

l)eriod,  and  at  the  second  j— ^  =         -   .  i/Vi3j^„;  while  the 

monev-value,  or  price,  of  the  individual  in  the  class  B  is  at  the 
first  period  y.\  and  at  the  second  yf:l^.     Now  ^  ~  •  y  '^^ i^ii^2  ' 

Vi^i   •  •  l/i^-^  '•      I   ^   •  '/  ^ ^/^i/^' '   '"^^^  ^^^^  geometric  mean    of  the 
A  «^ 

prices  of  the  individual  in  A  at  both  the  periods  and  the  geo- 
metric mean  of  the  prices  of  the  individual  in  B  at  both  the 
periods  are  both  y  V^/Jjyj'g.  AVhat  is  so  demonstrated  of  two 
classes,  can  be  demonstrated  of  any  number  of  classes. 

A  consequence  of  these  relations  is  that  if  we  suppose  the 
prices  of  these  individuals  iu  A  and  in  B,  etc.,  to  vary  at  a  uni- 
form rate  (that  is,  at  the  same  percentage,  compounding),  their 
prices  will  pass  the  geometric  mean,  the  same  in  all  cases,  at  the 
same  moment,  which  is  the  timal  half-way  point,  and  they  will 
then  be  actually  of  the  same  size  or  importance  (in  exchange- 
value),  and  at  any  other  moment  their  sizes  will  be  above  and 
below  this  in  corresponding  proportions.  These,  however,  are 
imaginary  or  ideal  relationships,  as  we  cannot  expect  prices  to 
vary  uniformly,  and  if  the  periods  be  contiguous,  the  price-vari- 
ations are  really  Avithin  each  period,  and  not  between  them. 
Still  they  are  the  relationshi])s  we  should  demand  when  we  con- 
ceive of  the  problem  in  its  ideal  conditions.  ♦ 

We  have,  then,  four  reasons  for  preferring  the  geometric 
mean  betsveen  the  total  money-values  at  the  two  periods, 
namely,  to  recapitulate  :  (1)  the  geometric  mean  avoids  an  iu- 


110    ARRANGEMENT    OF    PARTICULAR    EXCHANGE-VALUES 

consistency  into  which  the  arithmetic  mean  falls ;  (2)  it  alone  ^^ 
is  nnaifected  by  variations  or  differences  in  the  exchans^e-vahie 
of  money  ;  (3)  according'  to  it  alone  the  economic  individuals  in 
the  classes  have  inverted  relationships  at  each  of  the  periods, 
and  (4)  coincide  in  size  or  importance  at  the  timal  half-way 
point,  if  their  price-variations  are  supposed  to  be  at  a  uni- 
form rate. 

So  conceived,  the  economic  individual  in  every  class,  when 
the  mass-quantities  are  constant  over  both  periods,  is  composed 
at  both  periods  of  a  constant  mass-quantity,  which  varies  in  im- 
portance or  economic  size  (exchang:e-value),  in  such  a  way  that 
at  the  one  period  it  is  as  much  more  or  less  important  than  every 
other  as  at  the  other  period  it  is  less  or  more  important  than 
every  other,  so  that  altogether  it  has  the  same  importance  as 
every  other.  Xaturally  the  sizes  of  the  classes  are  according  to 
the  numbers  of  such  economic  individuals  they  contain. 

§  8.  III.  Lastly  we  have  to  consider  the  cases,  probably  the 
only  actual  ones,  when  both  the  mass-quantities  and  the  total 
exchange-values  of  the  classes  vary  from  the  one  period  to  the 
other.  Here  there  are  only  three,  or  at  most  four,  main  posi- 
tions that  claim  onr  serious  attention.  The  first  is  that  Ave 
should  take  the  smaller  total  exchange-value  of  every  class  at 
either  period  as  its  Aveight  in  the  comparison  of  the  two  periods. 
The  second  is  like  unto  this,  from  the  point  of  view  of  the  mass- 
quantities,  namely  that  we  should  take  only  the  smaller  mass- 
quantity  of  every  class  at  either  period,  and  then  treat  these 
quantities  as  we  would  treat  them  were  they  the  only  ones  at 
both  the  j>eriods,  taking  for  the  weight  of  every  class  the  geo- 
metric mean  between  the  total  money-values  of  these  reduced 
mass-quantities  at  the  prices  of  each  of  the  periods.  The  third 
is  that  we  should  treat  the  total  money-values  of  the  total  mass- 
quantities  at  both  periods  in  the  same  way  we  have  just  recom- 
mended for  treating  the  total  money- values  of  the  mass-quan- 
*  titles  that  are  constant  over  both  })eriods,  namely  by  taking  for 

^^  Consideration  of  the  harmonic  mean  is  here  omitted  because  no  chiim  has 
ever  been  put  forward  for  it,  and  no  reason  is  apparent  in  its  favor.  It  involves 
the  same  inconsistencies  and  difficulties,  inverted,  as  the  arithmetic. 


THE    qUESTIOX    OF    PERIODS  111 

the  weight  of  every  elass  the  jreonietrie  mean  between  the  class's 
fnll  total  nioney-valnes  at  both  ])criods.  Among  the  several 
suggestions  above  reviewed  for  treating  the  divergent  mass- 
quantities  only  one  has  any  good  claim  for  consideration.  We 
cannot  use  the  mass-quantities  of  each  period  separately,  in  the 
system  of  double  weighting  as  employed  by  Drobisch,  because 
that  ])erverts  the  problem,  as  we  shall  find  in  the  next  chapter. 
There  may  be  other  systems  of  double  weighting,  as  notably  one 
m vented  by  Professor  Lehr  ;  but  we  may  more  conveniently 
examine  these  later.  And  at  all  events,  so  long  as  we  consider 
the  question  of  averaging  price-variations,  instead  of  measuring 
variations  of  price-averages — a  distinction  which  will  be  made 
plainer  in  the  next  Chapter, — we  shall  need  to  choose  a  method 
of  single  weighting ;  and  the  single  system  itself  that  is  adopted 
will  be  the  basis  for  any  double  weighting  we  may  later  find 
proper  to  adopt.  To  take  the  mass-quantities  of  either  period 
alone  is  absurd,  as  neither  is  of  more  importance  than  the  other. 
T(  >  make  one  calculation  on  the  mass-quantities  of  the  one  period 
and  another  on  those  of  the  other,  and  then  to  take  the  (arith- 
metic) mean  between  the  two  results,  is  an  arbitrary  proceeding. 
The  only  position  hitherto  recommended  that  deserves  any  at- 
tention is  that  of  taking  the  (arithmetic)  means  between  the 
mass-quantities  at  the  two  periods,  or,  which  is  the  same  thing, 
the  sums  of  the  mass-quantities  at  both  the  periods,  and  treating 
these  as  if  they  were  mass-quantities  constant  over  both  the 
periods.'^  And  yet  a  question  will  arise  as  to  whether  the  geo- 
metric mean  is  not  better  here  also. 

The  first  of  these  methods  is  very  much  like  the  method  above 
discussed  of  taking  the  smaller  total  money-value  at  either  period 
when  the  mass-quantities  are  the  same  at  both  the  periods.  It 
shares  the  defects  of  that  inethod,  and  adds  to  them  another. 
This  is  that  if  we  happen  to  be  comparing  a  prosperous  period 
with  a  dull  period,  or  a  large  country  with  a  small  country,  the 
smaller  total  money-values  may  be  mostly,  and  even  all,  those 
of  the  dull  period,  or  of  the  small  country,  even  though  the  ex- 

i-*  This  and  the  preceding  method  prohably  yield  very  nearly  the  same  results. 
There  is  one  condition  in  which  their  results  always  agree  exactly.  This  is  in- 
dicated in  Appendix  C  IV   g  2  (.'!). 


112    ARRANGEMENT    OF    PARTICULAR    EXCHANGE-VALUES 

cbang'e- value  of  monev  may  be  the  same  at  l)(»tli  the  periods,  or 
in  both  the  countries,  and  the  total  money-values  correctly  repre- 
sent the  total  exchang-e- values.  For  this  added  defect,  however, 
a  correction  is  possil)le,  wliich  will  be  pointed  out  presently. 

The  second  has  more  to  recommend  it.  We  are  seeking  to 
measure  the  exchange-value  of  money  in  relation  to  other  classes 
of  things,  all  which  compose  a  whole.  If  all  these  classes  of 
things  consist  of  the  same  masses  at  both  })eriods,  the  material 
whole,  or  world,  which  tluy  com]>ose,  is  the  same,  or  simikir,  at 
botli  periods.  And  that  the  material  world  should  be  the  same, 
or  similar,  at  both  periods,  in  spite  of,  and  underneath,  the  vari- 
ations in  the  exchange-values  of  the  things  in  it,  has  the  ap- 
pearance of  being  a  postulate  of  simple  mensuration.  Now  by 
eliminating  all  the  surplus  mass-quantities  that  exist  at  either 
period  alone,  we  pick  out  all  those  mass-quantities  which  are 
common  to  both  the  periods,  so  that  a  material  world  composed 
of  these  mass-quantities  is  actually  the  same,  or  similar,  at  both 
periods. ^^  At  all  events  the  selection  of  these  mass-quantities 
seems  to  provide  an  answer  to  the  objection  advanced  by  Professor 
Sidgwick  and  l)y  Dr.  Wicksell.  There  is  also  another  argument, 
ap})arently  in  favor  of  this  position,  that  may  be  derived  from  a 
more  general  })osition,  which  here  deserves  to  be  noticed. 

§  9.  When  new  classes  of  commodities  appear  upon  the  scene 
in  the  course  of  our  comparisons,  the  appearance  of  these  in  no 
wise  affects  the  exchange-value  of  the  others.  At  first,  to  take 
a  simple  example,  our  money-unit  might  have  ])urchased  so  much 
of  one  kind  of  thing,  or  so  much  of  another ;  that  it  should 
later  be  able  to  purchase  the  same  amounts  of  these  things  and 
so  much  of  a  new  kind  of  thing,  widens  the  range  of  its  ex- 
changes, but  does  not  increase  its  power  over  other  things.  The 
change  is  extensive,  not  intensive.  Or,  viewed  in  another  way, 
at  first  the  money-unit  could  purchase  half  as  much  of  the  first 

'■''If  this  be  the  right  position,  then  tlie  treatment  ahove  recomuiemh'd  of  the 
special  cases  when  the  total  money-values  of  every  class  happen  to  remain  con- 
stant, or  to  vary  in  the  same  proportion,  has  to  be  abandoned.  For  then,  some  of 
the  prices  being  supposed  to  have  varied,  the  total  mass-quantities  must  have 
varied,  and  therefore,  according  to  the  present  method,  only  the  smaller  of 
these  total  mass-quantities,  and  their  reduced  total  money-values,  must  l>e  taken 
into  consideration. 


TJIE    QFESTIOX    OF    PERIODS  113 

and  lialfas  iiuich  df'tlic  second  kind  oi"  tliiniis  ;  and  later  it  will 
piirehase  a  third  of  the  first  and  a  third  of  the  second  and  a  third 
of  the  new  article.  Here  what  is  g-aincd  iu  extension,  is  lost  iu 
intension.  A  new  class  of  goods  adds  a  new  particular  exchange- 
value  to  the  general  exchange-value  of  everythmg  else  ;  but  this 
new  exchange-value  is  only  equal  to  each  of  the  old  ones,  and 
equal  to  the  general  exchange-value  (according  to  Pro])()sitiou 
VII.),  and  therefore  makes  no  change  in  any  of  those  magni- 
tudes. A  variation  in  anything's  general  exchange-value  can  be 
brought  about  only  by  a  variation  in  one  or  more  of  its  already 
existent  particular  exchange- values ;  and  such  a  variation  can 
take  place  only  relatively  to  a  class  of  objects  existing  at  both 
the  })eriods.  A  particular  exchange-value  relatively  to  a  class 
which  exists  only  at  one  of  the  periods  compared  cannot  have 
varied.  Consequently  a  class  of  things  is  not  to  be  counted  in 
the  one  period  when  it  is  not  counted  in  the  other. ^'^  And  in  a 
series  of  comparisons,  if  a  new  class  reaches  sufficient  imjiortance 
to  deserve  to  be  counted  at  a  certain  period,  in  comparing  this 
period  with  the  preceding,  iu  which  it  is  not  counted,  it  must 
still  be  neglected  ;  and  it  is  to  be  counted  only  in  the  comparison 
of  the  next  period  with  this  period,  and  thereafter. '^  And  in 
comparing  with  the  present  a  period  in  the  distant  past  we  must 
leave  out  the  classes  which  have  come  into  existence  since  and 
those  which  have  passed  out  of  existence,  comparing  only  those 
parts  of  each  world  which  are  conunon  to  both.  Here,  however, 
there  is  much  necessary  imperfection  in  the  comparison,  because 
some  classes  may  have  23assed  out  of  existence  only  recently,  and 
others  may  have  come  into  existence  soon  after  the  first  period, 
and  these  may  have  varied  in  the  intervening  periods.^^ 

1^  Thus  Fauveau  said  he  assumed  only  the  same  species  of  things  at  both  the 
periods,  B.  54,  p.  355. — This  principle  may  be  violated  in  Drobisch's  method.  In- 
difference to  its  violation  is  shown  bj'  Lindsay,  B.  114,  p.  26.  The  need  of  ob- 
serving it  is  treated  as  a  defect  by  Nicholson,  B.  i>4,  ff.  313-315. 

^■'Attention  was  called  to  this  problem  by  Sidgwick,  who  thought  it  one  of 
great  difliculty,  B.  50,  p.  68.  It  was  solved  by  Marshall,  B.  93,  p.  373  n.  The 
solution  was  adopted  by  the  British  Association  Committee,  First  Report,  B.  99, 
p.  250. 

^*  Hence  in  such  cases  true  comparisons  can  l)e  made  only  by  passing  through 
all  the  intervening  periods.  Where  this  is  impossible,  a  true  comparison  is  im- 
possible. 

8 


114    ARRANGEMENT    OF    PARTICULAR    EXCHANGE-VALUES 

The  knowledge  gamed  by  this  investigation  into  the  want  of 
influence  of  new  classes  it  may  be  well  now  to  state  in  preposi- 
tional form.  From  this  investigation  (and  really  as  a  corollary 
to  Proposition  VII.)  we  perceive  that  the  height  of  the  general 
exchange-value  of  anything  is  in  no  wise  determined  by  the  num- 
ber of  its  particular  exchange-values,  but  solely  by  their  (common) 
height  (Proposition  XLI.).  And  so,  there  being  no  variation  of 
exchange-values  (or  of  jprices)^  the  appearance  of  a  new  class  of 
things,  with  any  exchange-value  whatsoever,  has  no  influence  upon 
the  height  of  the  exchange-value  of  anything  already  existing,  nor 
can  it  have  any  such  influence  until  it  has  existed  over  at  least 
two  periods,  thus  having  time  to  remain  constant  or  to  vary  ; 
nor  can  the  disappearance  of  an  old  class  have  any  such  influence, 
no  matter  tvhat  ivas  its  exchange-value  at  the  time  ivhen  it  disap- 
peared (Proposition  XLII.). 

Now  what  is  thus  shown  to  be  true  of  the  appearance  or  dis- 
appearance of  a  class,  it  might  be  argued,  is  also  true  of  the  ap- 
pearance or  disappearance  of  individual  (physical  or  material) 
things  in  old  classes.  It  might  be  argued  that  just  as  we  must 
count  only  the  classes  of  things  which  are  common  to  botli  the 
worlds  compared,  so  in  each  class  we  must  count  only  the  (ma- 
terial) individuals  which  are  common  to  them  both.  The  reason 
why  the  classes  which  appear  only  at  one  of  the  periods  cannot 
be  counted  is  that  such  classes  have  neither  varied  nor  remamed 
constant  in  exchange-value  and  money  has  neither  varied  nor 
remained  constant  in  exchange-value  in  them.  The  same  reason 
might  seem  to  apply  to  (material)  individuals  which  are  present  at 
only  one  of  the  periods.  These  have  neither  varied  nor  remained 
constant  in  exchange-value,  nor  has  money  either  varied  or  re- 
mained constant  in  exchange-value  in  them.  Therefore  these 
single-period  individuals,  it  might  seem,  should  be  neglected  in 
our  calculations. 

§  10.  What  has  above  been  said  of  classes  is  true  of  (material) 
individuals  as  follows.  The  height  of  tlie  general  exchange- 
value  of  anything  is  not  determined  by  the  number  of  its  in- 
dividual exchange-values,  but  solely  by^  their  {common)  height 
(Proposition   XLIII.).     Hence,  if  there  are  no  variations  of 


THE    (QUESTION    OF    PERIODS  115 

prices,  or  of  the  exchange-values  of  thinos  in  money,  and  of 
money's  inverse  exchange-vahies  in  all  other  things,  so  that  the 
general  exchange-value  of  money  is  constant  (according  to  Prop- 
osition XXVII.),  then  the  coming  into  existence  of  new  indi- 
viduals, or  the  ])assing  out  of  existence  of  old  ones,  with  the 
constant  prices  of  the  other  individuals  in  the  same  classes,  does 
not  aifect  the  height  of  any  particular  exchange- value  of  money, 
and  consequently  not  the  height  of  its  general  exchange- value. '^ 
In  other  M'ords,  a  change  in  the  size  of  any  classes,  without  a 
variation  In  any  exchange-values  (^or  prices),  does  not  cause  a 
variation  in  anytJthufs  general  r.vrhange-value  (Proposition 
XLIV.)."''  This  means  that  wlien  there  are  no  variations  of  ex- 
change-values (or  of  prices),  the  weighting  of  the  classes  is  indif- 
ferent. Hence  it  is  indiflPerent,  in  this  case,  whether  we  neglect 
the  siu-plus  individuals  or  not.  It  may  be  added  that  also  If  (dl 
prices  vary  in  the  same  proportion,  changes  in  the  sizes  of  any 
classes  do  not  affect  the  variation  in  the  exchange-value  of  money, 
which,  is  t/ie  inverse  of  this  common  price-varirdion  (Proposition 
XLV.);^^  and  consequently  the  weighting  of  the  classes  is  indif- 
ferent also  in  this  case. 

But  when  there  is  irregular  variation  of  some  exchange-values, 
or  prices,  between  t^\•o  periods,  then  the  weighting  is  very  im- 
portant. And  now  also  it  is  important  whether  any  of  the 
classes  ^^  has  augmented  or  diminished  or  remained  constant  in 
size.  That  allowance  should  be  made  for  such  changes  in  total 
exchange-value  (or  in  total  money-value)  size,  we  have  already 

i^For  at  a  given  moment  tinal  utilities  are,  to  the  people  at  large,  according 
to  prices.  At  a  given  moment,  then,  prices  being  given,  if  people  spend  their 
money  more  on  one  class  than  on  another,  or  reversely,  this  only  shows  that  they 
are  choosing  the  final  utilities  in  a  diiferent  manner,  not  that  they  are  getting 
more  final  utility  by  spending  their  money  in  one  way  than  in  another.  There- 
fore at  two  moments,  or  periods,  all  prices  still  being  the  same,  if  people  actually 
do  spend  their  money  differently,  this  shows  nothing  more  or  less  than  was  shown 
in  the  previous  supposition. 

20  This  Proposition  flows  directly  from  Proposition  XX  VII.,  wliich  is  uncon- 
ditional. 

2^  This  Proposition  flows  directly  from  Proposition  XVII.,  wiiich  is  uncon- 
ditional. Compare  also  Propositions  XXXV.  and  XXXVI.,  from  which  a  simi- 
lar corollary  may  be  tlrawn  concerning  the  weighting,  in  all  cases,  of  any  class 
whose  price  varies  inversely  to  the  exchange- value  of  money — either  in  all  things 
or  in  all  other  things,  according  to  the  measurement  that  is  being  made. 

2  2  Except  as  indicated  in  the  preceding  note. 


116    ARRAXGEMEXT    OF    PARTICULAR    EXCHAXGE-YALUES 

conceded.  Then  Avhy  should  not  allowance  ])e  made  also  for 
changes  m  material  size  ?  The  economic  individual  may  change 
in  exchange-value  size  between  the  two  periods.  Then  why 
should  not  the  economic  individual  change  also  in  material  size  ? 
It  is  true  that  it  is  only  the  lesser  mass-quantity  of  any  class 
existing  at  either  period,  or  the  mass-<|uantity  common  to  both 
periods,  that  has  varied  or  remained  constant  in  exchange-value. 
Yet  money  has  varied  or  remained  constant  in  exchange-value 
in  a  changing  mass-quantity  of  this  class,  just  as  it  has  varied 
or  remained  constant  in  exchange- value  m  a  changing  exchange- 
value  (quantity.  AVhen  a  class  is  wholly  absent  from  the  one 
})eriod,  though  ])reseut  at  the  other,  we  have  no  price-quotation 
of  it  at  all  at  the  one  period,  and  so  it  can  have  no  variation  or 
constancy  in  itfe  price  or  exchange-value.^^  But  when  merely  a 
certain  material  quantity  of  a  class  is  absent  from  the  one  i)eriod, 
vet,  as  its  similar  mates  are  present  at  both  ])eriods,  Ave  do  know 
the  variation  or  constancy  of  money  in  exchange-value  in  this 
changing  class  of  things. 

§  11.  A  })ractical  objection  also  exists  against  taking  only 
the  lesser  mass-quantities  at  either  period,  like  that  brought 
against  taking  only  the  lesser  total  money-values  of  the  classes 
at  either  ])eriod.  This  is  that  if  we  are  compa^ring  a  prosperous 
period  with  a  dull  period,  or  a  large  countiy  with  a  small 
country,  the  mass-quantities  might  be  only  those  of  the  small 
country — which  would  be  treating  the  small  country  as  if  its 
commodities  Avere  more  important  than  those  of  the  large  country, 
— or  only  those  of  the  dull  ])eriod,  which  might  in  one  case  be 
the  earlier,  and  in  another  the  later  period,  so  that  in  the  former 
case  we  should  be  using  the  mass-quantities  of  the  first  period  in 
the  comparison  of  the  two  periods,  and  in  the  later  the  mass- 
quantities  of  the  second  period,  the  only  reason  for  the  diiierence 
in  the  choice  of  the  period  being  that  more  importance  is  attached 
to  the  conditions  in  the  dull  period  than  to  those  in  the  pros- 
perous period, — which  is  absurd. 

Still  a  correction  of  this  defect  in   this  method — and  at  the 

2  3  For  surely  we  eannot  supply  what  wduUI  have  been  its  price  had  it  existed, 
as  suggested  by  Nicholson,  B.  !t4,  p.  :J14. 


THE    QUESTION    OF    PERIODS  117 

samo  time  of  the  similar  defect  in  the  first-noticed  method — 
conld  be  made  in  the  Ibllowiug  manner.  Finding  the  total 
money -expenditure  at  each  of  the  periods,  or  in  each  of  the 
countries,  reduce  the  expenditure  in  the  one  to  the  expenditure 
in  the  other,  and  reduce  all  the  mass-quantities  in  the  same  pro- 
portion ;  then  take  the  smaller  of  these  mass-(]uantities  at  either 
period,  or  country,  and  operate  as  before — or,  in  the  other 
method,  take  the  smaller  of  the  total  money-values  so  reduced. 
This  correction,  however,  generally  needs  a  further  correction. 
For  it  involves  that  the  total  money-expenditures  so  compared 
shall  be  of  the  same  exchange-value ;  and,  therefore,  it  can  be 
safely  employed  only  Avhen  the  exchange-value  of  money  is  the 
same  at  both  the  periods,  or  in  both  the  countries.  When  this 
is  not  the  case,  or  not  known  to  be  the  case,  as  always  when  we 
are  attempting  to  measure  the  exchange-value  of  money,  the 
only  further  correction  can  be  by  an  uncertain  method  of  a])- 
proach,  like  that  previously  noticed.  For  perhaps  the  truth 
may  be  reached  by  taking  the  first  result  of  tliis  methcjd  as 
representing  the  variation  in  the  exchange-value  of  money,  and 
then  again  reducing  the  total  money-expenditures  to  the  same 
total  exchange-value  expenditure  on  the  assumption  of  this 
variation  being  correct,  and  repeating  the  operation  until  the 
result  tallies  with  the  last  assumption. 

Another  of  the  above  suggested  methods,  namely  that  of  tak- 
ing the  arithmetic  mean  between  the  mass-quantities  at  each 
period,  or,  which  is  the  same  thing,  the  total  mass-quantities 
over  both  the  periods,  has  no  theoretical  reason  in  its  favor ;  for 
it  is  not  the  total  mass-quantities  during  both  the  periods,  or  the 
halves  of  these,  that  have  varied  from  the  one  period  to  the 
other,  since  more  than  half  of  them  may  be  within  one  of  the 
periods. 

Moreover  this  method  stands  in  need  of  the  same  sort  of  cor- 
rection as  the  preceding.  For  between  a  small  country  and  a 
large  country,  or  between  a  dull  period  and  a  pros}>erous  ])eriod, 
the  arithmetic  mean  of  the  mass-quantities  Avould  l)e  more  in- 
fluenced by  the  conditions  in  the  large  country,  or  in  the  pros- 
perous period — the    infiucncc   here    being   on    the    o])posite  side 


118    ARRANGEMENT    OF    PARTICULAR    EXCHANGE- VALUES 

from  what  it  was  on  iu  the  precedmg  cases.  The  correction  is  to 
be  made  in  the  same  way  as  before,  namely  by  redncing  the  mass- 
fjuantities,  before  averaging  them,  in  the  same  proportion,  so 
that  their  grand  total  exchange-values — not  merely  money- 
values — shall  be  the  same  at  each  period  ;  all  which  again  in- 
volves the  need  of  approach  through  many  assumptions  and 
repetitions,  so  that,  from  being  a  simple  and  convenient  method, 
it  becomes  a  very  laborious  one. 

A  simpler  means  of  avoiding  at  once  the  difficulty  arising 
from  differences  in  sizes  of  the  countries  or  in  prosperity  of  the 
periods  and  that  arising  from  variation  in  the  exchange-value 
of  money,  consists  in  eschewing  the  arithmetic  mean  of  the  mass- 
quantities  and  substituting  the  geometric. 

It  may  be,  however,  that  in  spite  of  its  theoretical  insufficiency, 
the  method  of  taking  merely  the  aritlunetic  means  of  the  mass- 
quantities,  uncorrected,  may  in  ordinary  cases  yield  results  so 
near  the  truth  that  it  may  be  preferable  to  any  of  the  other 
methods  thus  far  examined,  and  on  account  of  its  greater  con- 
venience may  be  even  preferred  to  the  truer  methods  which  we 
shall  later  discover.  But  the  practical  merits  of  these  methods 
it  is  impossible  to  examine  here.  We  shall  examine  them 
later  in  connection  with  the  averages  that  are  to  be  drawn  of 
the  price  variations. 

§  13.  The  suggested  method  above  set  down  as  the  third  still 
remains  for  examination.  This  is  simply  to  draw  the  geometric 
mean  of  the  full  weights  of  every  class  at  both  periods,  that  is, 
of  the  total  money-values  (which  indicate  the  relative  total  ex- 
change-values) of  the  classes  at  both  the  periods,  without  having 
any  more  concern  for  the  changes  in  the  mass-quantities  of  the 
classes  than  foi-  the  changes  in  their  total  exchange-values.  In 
the  very  first  case  considered,  where  the  total  money-value  of 
every  class  hapjiens  to  be  constant  over  both  periods,  or  to  change 
in  the  same  pro])ortion,  so  that  the  relative  sizes  of  the  classes 
remain  unchanged,  we  have  seen  that  the  most  natural  treatment 
is  to  take  these  relative  money-values  as  the  weights  of  the 
classes.  Then,  in  the  simplest  cases,  the  economic  individual  is 
a  constant  money-value  at  both  periods,  but  is  a  changing  mass- 


THE  (2Up:stiox  of  pp:riods  119 

quantity.  In  tlie  .'-^c'coikI  general  case,  where  the  nias.s-quautities 
in  all  the  classes  ha])pen  to  be  constant  over  both  periods,  the 
economic  individual  Ave  have  unhesitatingly  found  to  be  a  con- 
stiint  mass-quantity  at  both  periods,  but  a  changing  money-value 
(or  exchange-value).  Neither  of  these  cases  is  likely  ever  to 
occur  in  actuality.  Now,  when  both  the  total  money-values  and 
the  total  mass-quantities  in  every  class  are  various  at  both  the 
periods,  it  seems  most  natural,  and  not  improper,  to  combine 
both  these  conceptions,  and  to  have  for  our  economic  individual 
one  changing  both  in  money-value  (or  exchange-value)  and  in 
mass-quantity.-*  As  before,  the  nature  of  this  individual  may 
be  best  understood  by  aid  of  an  example. 

Suppose  at  the  first  period  we  have  100  pomids  of  A  at  $1.00 
apiece  and  100  poimds  of  B  at  $1.00  apiece,  and  at  the  second 
period  50  pounds  of  A  at  $1.9()  apiece  and  1,000  pounds  of  B 
at  $0.04  apiece.  The  total  money-values  of  the  class  A  are 
$100  at  the  first  period  and  $98  at  the  second,  and  those  of  the 
class  B  $100  and  $40  respectively.  The  relative  sizes  of  these 
classes,  therefore,  according  to  this  method  of  measurement,  are 
1/9800  =  98.995  for  the  class  A,  and  1/4000  =  63.246  for  the 
class  B.     Or  if  the  latter  be  taken  as  one  unit,  the  weight  for  A 

.    98.995       ^  ^n~(         |9800       ,/ir^S\       rp,       .        .,      • 

IS  =  1.56ol  =      =  V  2Ao  I  •     Theretore,  the  m- 

63.246  V     \4000  J 

.    -,    ^    100 
dividual  in  the  class  A  consists  at  the  first  period  ot  :p^  -  = 

,         n  50 

63.89   pounds,   worth   $63.89,  and  at  the  second  of  YrW  ^ 

31.945  pounds,  worth  $62.61  ;  and  the  individual  in  the  class 

B  consists  at  the  first  period  of  100  pounds,  worth  $100,  and  at 

100        62.61 
the  second  of  1000  pounds,  worth  $40.     But  wo^^^  =      «q 

=  1.565,  that  is,  the  individual  in  B  is  1.565  times  more  ^valu- 
able than  the  individual  in  A  at  the  first  period,  and  at  the  second 
period  the  individual  in  A  is  1.565  times  more  valuable  than  the 
individual  in  B,  although  the  individual  in  A  has  contracted  by 

24  If  this  method  be  admitted,  both  those  methods  are  to  be  retained,  and  em- 
ployed when  the  conditions  are  met.  This  method  is  comprehensive,  and  includes 
both  those. 


120    ARRANGEMENT    OF    PARTICULAR    EXCHANGE- VALUES 

half,  and  the  iudividual  in  B  has  grown  tenfold,  in  mass-size. 
Thns  in  spite  of  the  changes  in  their  mass-sizes  the  individuals, 
so  obtained,  in  the  classes  have  inverse  relations  of  money-value 
(and  consequently  of  exchange-value)  at  each  of  the  two  periods, 
and  therefore  are  equally  valuable,  or  equivalent,  over  the  two 
periods  together.  Also  l/ 6:3. 8 9 ^^6 2761  =  l/lOO  x  40  =  63.- 
246,  that  is,  the  geometric  mean  of  the  prices  of  the  individual 
in  A  is  the  same  as  the  geometric  mean  of  the  prices  of  tlie  in- 
dividual in  B,  which  means  that  at  the  half-way  moment  between 
the  two  })eriods,  the  prices  being  supposed  to  vary  at  a  uniform 
rate,  the  prices  of  the  two  individuals  are  the  same,  so  that  then 
these  two  individuals  are  actually  equivalent. 

That  these  two  relationships  are  universal,  can  be  demon- 
strated as  easily  as  before  in  the  case  when  the  mass-quantities 
were  the  same  at  both  periods  (see  above,  §  7).  We  have 
only  to  distinguish  x  into  x^  and  .r.„  and   //  into  y^  and    //.„  and 

then  the  price  of  the  individual  in  the  class  A  becomes      l-^-^. 

l/^j2/2/?i/?2  ^^  *^^^  ^^"^^  period  and  ^j   ^—  •  ^'^ViyJ^i^-i  ^^  *^®  ^^^~ 

ond,  and  everything  works  out  as  before. 

Variations  in  the  exchange- value  of  money  have  no  influence 
to  derange  the  weights  here,  just  as  they  have  none  in  the  cases 
where  the  mass-quantities  are  the  same  at  both  periods,  every 
weight  being  affected  in  the  same  proportion  here  as  there. 
Nor  does  a  difference  in  the  sizes  of  the  countries,  or  in  the 
prosperity  of  the  periods,  have  any  such  deranging  influence. 
The  geometric  mean  with  even  weighting  being  employed,  equal 
weight  is  really  attached  to  the  weights  of  each  period.  There 
is  no  need  of  correction.  This  method  of  weighting,  then,  is 
not  only  the  best,  but  even  the  simplest. 

At  all  events  this  method  seems  to  give  us  the  true  concej)- 
tion  of  the  economic  individual  in  a  comparison  between  two 
periods.  That  the  variations  of  the  prices  of  the  classes,  be- 
tween two  j)eri(^ds,  should  be  weighted  according  to  the  relative 
numbers  of  such  individuals  the  classes  contain  over  the  two 
periods,  would  seem  to  be  propci*.      Or  if  we  average  prices  at 


WAGES    EXCLUDED  121 

each  period  separately,  ])reparat()rv  to  comj^arino:  the  averages, 
it  woiikl  seem  proper  to  make  use  of  the  mimhers  of  these  indi- 
viduals the  classes  contain  at  each  jx'riod,  determined  with  refer- 
ence to  the  other  period  with  which  it  is  compared. 

§  14.  The  subject  has  to  be  left  in  this  somewhat  unsatis- 
factory state  until  we  have  examined  the  question  of  the  aver- 
ages in  which  these  systems  of  weighting  are  to  be  employed. 
One  system  of  weighting  may  })erhaps  be  found  suitable  for  one 
kind  of  averaging,  and  not  for  another.  And  one  kind  of 
weighting  with  one  kind  of  averaging  may  be  found  to  yield 
exactly,  or  very  nearly,  the  same  results  as  another  kind  of 
weighting  with  another  kind  of  averaging.  Hence  our  ch(»iee 
may  be  more  narrowed  than  it  is  at  ])resent. 

So  far  we  have  been  attempting,  without  complete  success  as 
yet,  to  reach  the  theoretically  exact  position.  To  attain  to  this 
position  itself  would,  however,  be  of  little  service,  unless  we 
have  very  exact  data  to  apply  the  theoretically  correct  method 
to.  Hence  the  theoretically  correct  method  of  weighting  will 
be  serviceable  onlv  for  measurino-  the  course  of  the  exchange- 
value  of  money  during  present  time,  as  we  advance  into  the 
future.  For  reviewing  what  is  already  past,  it  is  hopeless  to 
expect  to  find  data  sufficient  to  justify  us  in  making  use  of  any 
but  a  rough  and  ready  ^veighting,  the  same  for  many  consecutive 
periods  together,  the  finer  shades  of  diiference  from  year  to  year 
being  untraceable.  This  weighting,  the  same  for  many  periods, 
must  either  be  some  general  average  of  the  relative  importance 
of  the  classes  over  these  periods,  dv  be  according  to  some  gen- 
eral average  of  the  mass-quantities  in  the  classes  over  these 
periods.  Which  of  these  methods  is  the  better,  and  of  what 
sort  the  average  ought  to  be,  we  must  postpone  examining  till 
M'e  have  revicAved  the  subject  of  weighting  in  connection  witli 
the  subject  of  averaging  the  price  variations. 

YI. 

§  1.  One  more  item  remains,  about  which  there  has  been  dis- 
pute. Many  economists  have  maintained  that  among  the  prices 
of  things  which  are  to  be  counted   in  measuring  the  exchange- 


122        SELECTION    OF    PARTICULAR    EXCHAXGE-VALUES 

value  of  money  we  should  include  also  the  '^  price  of  labor," 
namely,  wao-es  and  salaries.' 

No  opinion  could  be  more  erroneous. 

§  2.  In  the  first  place  labor  has  no  exchange- value.  Labor 
is  not  a  possessible  thing  :  it  does  not  pass  from  one  owner  to  an- 
other ;  it  is  not  exchanged  for  anything  else.  What !  it  may  be 
said,  does  not  the  employer  pay  money  for  labor,  and  does  not 
tlie  laborer  get  money  for  his  labor?  By  no  means.  The  es- 
sence of  the  contract  between  a  manufacturer  and  his  employees 
is  that  the  former  shall  put  materials  and  machines  in  the  hands 
of  the  latter  and  shall  take  the  products  which  they  shall  make 
— he  buys  from  them  the  improvements  they  make,  they  sell  to 
him  those  improvements.'  Even  in  domestic  service  what  the 
employer  pays  for  is  the  product  of  labor  : — the  charwoman  is 

^  Wages  were  included  in  the  lists,  with  the  prices  of  commodities,  for  calcu- 
lating variations  in  the  "  value  "  of  money,  by  Dutot,  Evelyn  and  Young.  Adam 
Smith  found  the  "value  "  of  silver  at  different  epochs  by  considering  the  sums 
needed  to  purchase  "the  same  quantity  of  labor  and  commodities,"  oj).  cit.,  p. 
101.  (His  inexactness  is  shown  by  his  making  the  same  measurement  by  "the 
quantity  of  labor  which  any  particular  quantity  of  them  [gold  and  silver]  can 
purchase  or  command,  or  the  quantity  of  other  goods  which  it  will  exchange  for," 
p.  14.)  McCulloch  :  to  be  constant  in  "  exchangeable  value"  a  thing  must  "  at 
all  times  exchange  for,  or  purchase,  the  same  quantity  of  all  other  commodities 
and  labor,"  Political  economy,  p.  213  (and  in  Note  to  Wealth  of  nations,  p.  439). 
Roscher  gives  a  truly  German  reason  for  assigning  "an  important  place  "  in  the 
lists  to  daily  wages  :  "  The  desire  to  exert  influence  upon  other  men  and  to  be 
prominent  socially  is  a  very  universal  one  ;  and  of  its  attainability  there  is  no 
better  sign  than  the  power  of  disposal  over  many  days  of  labor,"  B.  32,  §  129. 
That  wages  sliould  be  counted  along  with  commodities  has  been  of  late  asserted 
by  Martin,  op.  cit.,  p.  626;  Nasse,  Die  Wahrungsfrage  in  Deutschland,  Preus- 
sische  Jahrbiicher,  1885,  p.  313 ;  Giffen,  B.  44,  pp.  127,  128,  B.  45,  q.  780 ;  New- 
comb,  B.  76,  p.  212 ;  Wasserab,  B.  105,  p.  75 ;  G.  P.  Osborne,  Principles  of  econ- 
omics, Cincinnati  1893,  p.  332  ;  Lindsay,  B.  114,  p.  332  ;  G.  II.  Dick,  International 
bullion  money,  London  1894,  p.  3  ;  Wiebe,  B.  124,  pp.  168-169  ;  Edgeworth,  B. 
65,  p.  386 ;  A.  M.  Hyde,  Gold,  labor  and  commodities  as  standards  of  value, 
Journal  of  Political  Economy,  Chicago,  Dec.  1897,  p.  97;  Parsons,  B.  136,  pp.  IV., 
97,  115,  128. 

-  The  diSiculty  in  carrying  out  the  ordinary  opinion  is  well  shown  by  James 
Mill  in  his  Elements  of  2}olitical  economy  (2d  ed.,  1824).  He  starts  out  by  say- 
ing; "  The  laborer  who  receives  wages  sells  his  labor.  .  .  .  The  manufacturer  wlio 
pays  these  wages  buys  the  labor,"  p.  21 ;  and  later  says  that  when  a  capitalist  pro- 
vides raw  materials  and  tools,  and  the  laborer  works  up  the  product,  this  "  be- 
longs to  the  laborer  and  capitalist  together,"  but  that,  "when  the  share  of  the 
commodity  which  belongs  to  tiie  laborer  has  been  all  received  in  the  shape  of 
wages  [paid  in  advance  of  the  sale  of  the  product],  the  commodity  itself  belongs 
to  the  capitalist,  he  having  in  reality,  bought  the  siiare  of  the  laborer  and  paid 
for  it  in  advance,"  pp.  40-41.     On  p.  90  he  makes  both  these  statements ! 


WAGES    EXCLUDED  123 

pakl  for  clean  windows  and  floors  made  out  of  dirty  windows 
and  floors,  the  cook  for  cooked  food  in  the  place  of  raw  food,  the 
waiter  for  food  produced  on  the  table  from  food  produced  in  the 
kitchen,  the  coachman  for  a  moving  carriage  instead  of  a  station- 
ary one.'  Emj)l()yers  do  not  want  labor — they  would  ])av  much 
more  if  they  could  be  served,  like  Psyche  in  Cu[)id's  bower,  Avith 
the  hands  of  invisible  spirits.  What  they  want,  and  what  they 
pay  for,  is  something  which  they  cannot  get  without  paying  some 
one  to  produce  it  for  them.^  And  what  laborers  give  in  return 
for  their  hire  is  not  the  labor  which  nobody  Avauts,  but  the  ])ro- 
ducts  of  that  labor.^ 

§  3.  But  this  is  not  the  principal  reason,  although  the  princi- 
pal reason  flows  from  this.  Labor  and  material  things  being 
uninterchangeable,  the  wages  of  labor  and  the  prices  of  com- 
modities are  categorically  distinct.  To  put  them  in  the  same  list 
is  to  try  to  mix  oil  and  Avater.     AVages  belong  to  another  list. 

^  Some  writers  have  admitteil  that  the  wages  of  "  productive  labor  "  (wages 
paid  for  the  productioQ  of  goods  that  are  to  be  sold)  areuot  to  be  counted  because 
they  are  a  factor  eiiteriug  into  the  prices  of  the  goods  produced,  and  therefore, 
being  already  counted  in  those,  they  would  be  counted  twice  ;  but  claim  that  the 
wages  of  so-called  "  unproductive  labor"  (of  domestics,  etc.,  whose  products  are 
not  sold)  ought  to  be  counted  :— Edgeworth,  B.  59,  p.  266  n.  ;  Marshall,  B.  93,  p. 
372;  Nicholson,  B.  94,  p.  324;  the  British  Association  Committee,  First  Report, 
B.  99,  p.  249 ;  H.  J.  Davenport,  Outlines  of  economic  theory,  New  York  1896,  p. 
227;  Wicksell,  B.  139,  p.  18.  Here  a  distinction  is  to  be  drawn.  The  wages  of 
domestics,  etc.,  contribute  only  to  the  retail  prices  we  pay  for  what  we  consume. 
Therefore  if  retail  prices  are  being  used  in  our  lists,  these  wages  would  properly 
belong  there.  But  retail  prices  are  usually  excluded,  for  reasons  already  ex- 
plained.   Hence  the  wages  even  of  domestics  should  be  excluded. 

*So  Fonda,  B.  127,  pp.  14-15,  and  Davenport,  op.  cit.,  p.  .53.  Cf.  Aristotle, 
who  said  we  should  have  no  need  of  workmen  if  shuttles  moved  themselves. 
Politics,  I.,  2,  5. — On  the  other  hand,  that  labor  is  the  only  thing  we  pay  for  has 
been  asserted  by  W.  D.  Wilson,  First  principles  of  political  economy,  Phila- 
delphia 1882,  p.  100. 

5  The  products  of  labor  for  which  we  exchange  money  may  l)e  immaterial 
things — dramatic  scenes,  music,  etc.  (pleasureable  sensations  and  thoughts),  or 
states  and  conditions  (safety,  health,  etc.) — in  short  services,  if  by  this  term  l)e 
meant,  not  actions,  but  the  effects  of  actions.  We  do  not  pay  musicians  for  play- 
ing, but  for  music  ;  nor  lawyers  for  pleading,  but  for  the  results  of  their  plead- 
ing,— often  paying,  however,  for  a  chance  of  getting  what  we  do  not  get.  Now 
immaterial  products  ought,  theoretically,  to  l)e  counted  in  the  lists — we  might 
count  theatrical  tickets,  fees,  etc.  The  sole  reason  why  they  are  to  be  excluded 
is  the  practical  one  already  applied  to  many  material  products,  that  their 
qualities  are  so  diverse,  the  individual  products  under  the  same  names  so  various, 
that  they  do  not  form  homogeneous  classes.  Yet  one  of  them,  transportation  of 
persons,  we  have  seen  to  deserve  to  be  counted. 


124        SELECTION    OF    PARTICULAR    EXCHANGE- VALUES 

Labor  is  the  cost  at  wliicli  tlie  majority  of  mankind  procure 
the  commodities  whose  prices  go  into  the  list  drawn  up  for 
measuring  the  exchange- value  of  money.  According  as  is  the 
labor  which  it  costs  a  man  to  produce  a  certain  thing,  so  is  the 
cost-value  of  the  thing  to  him.  According  to  the  average  labor 
which  it  costs  the  producing  part  of  mankind  to  produce  things 
of  a  certain  class,  so  is  the  average  cost-value  of  that  class  of 
things  in  general.  The  cost-value  of  a  thing  is  not  necessarily 
according  to  the  labor  merely  of  producing  the  thing  physically  ; 
for  things  are  carried  to  places  where  they  are  not  so  produced, 
and  the  economic  producticm — offering  in  the  market — of  them 
there  includes  the  labor  of  trausportatation.  The  majority  of 
mankind,  however,  do  not  produce  but  a  small  part  of  the  things 
tliey  consume,  and  so  are  more  interested  in  the  cost  oi  procuring 
things,  which  is  according  to  the  cost  of  producing  other  things, 
material  or  immaterial,  and  the  rate  at  which  these  can  be  ex- 
changed for  those.  Here  the  idea  of  cost-value  in  the  narrow 
sense  of  cost  of  production  proper,  passes  over  mto  a  wider  sense 
of  cost  of  acquisition,  in  which  it  becomes  identical  in  magnitude 
with  esteem-value.  If  a  man  gets  a  dollar  a  day  for  what  he 
produces,  whether  in  wages  or  otherwise,  then  anything  the  price 
of  which  is  a  dollar  costs  him  a  day's  work.  Such  a  thing  has 
not  to  him  the  exchange-value  of  a  day's  work  ;  for  he  does  not 
exchange  the  day's  work  for  it,  and  if  he  does  not  get  it,  through 
not  working  for  it,  he  has  lost  his  day's  work  as  well  as  the 
thing — he  has  idleness  without  the  thing  in  place  of  the  thing 
witli  work,  so  that  if  there  is  any  exchange,  it  is  exchange  of 
idleness  for  the  thing,  and  the  thing  should  be  said  to  have  the 
exchange-value,  not  of  a  day's  work,  but  of  a  day's  idleness. 
Nothing,  however,  is  reached  along  this  line  of  reasoning.  There 
is  properly  no  exchange,  but  there  is  cost."     There  is  comparison 

'^  The  conception  of  Turgot  and  Adam  Smith  that  labor  is  thereal  jjr/cewepay 
for  what  we  get  is  a  misuse  of  terms,  as  it  confounds  "price"  witli  "cost." 
Price,  in  a  wide  and  in  itself  not  very  proper  sense,  is  the  thing  which  one  man 
gives  to  another  in  exchange  for  what  the  other  gives  to  him.  Cost  is  what  we 
give  up,  without  anybody  else  getting  it.  If  a  man  who  owns  a  ton  of  coal  gives 
it  to  another  in  exchange  for  a  ton  of  ii-on,  the  ton  of  coal  is,  in  this  wide  sense,  the 
price  of  the  ton  of  iron  (and  the  exchange-value  of  the  ton  of  iron  in  coal  is  meas- 
ured by  this  amount).     If  a  man  burns  up  a  ton  of  coal  in  smelting  a  ton  of  iron, 


\vA(iKs  excludi<:d  125 

of  the  pleasure  in  procuriiio-  tlie  thing  with  the  di,s])leasure  of 
doing  the  work.  This  comparison  is  the  essence  of  esteeni- 
vahie,  which  permits  a  certain  amount  of  cost-value  to  things, 
and  no  more.  This  feature  in  esteem-value  shows  itself  again 
in  the  comparison  of  the  ])leasure  of  possessing  one  thing  with 
the  displeasure  of  abandoning  the  possession  and  use  of  another, 
or  of  renounehig  possession  and  use  of  othci-  things  also  pro- 
curable in  the  place  of  the  thing  obtained  ;  which  coni])arison 
regulates  the  exchange- values  of  things. 

§  4.  Now  it  is  desirable  that  we  should  be  able  to  measure 
the  cost-value  of  things  in  both  the  narrow  and  the  wide  sense. 
Industrial  progress  consists  in  cheapening  the  cost-value  of 
things,  and  it  is  Avell  to  know  whether  progress  is  being  made, 
and  how  rapidly.  But  the  trouble  of  measuring  the  cost-value 
of  everything  separately  might  be  saved  by  measuring  the  cost- 
value  of  money,  provided  this  be  its  cost-value  in  the  wide 
sense,  identical  M^th  esteem-value.  For  if  we  shoidd  find,  for 
instance,  that  this  cost-value  of  money  has  been  constant,  then 
we  should  know  that,  as  the  price  of  anything  varies,  the  cost- 
value,  at  least  in  the  wide  sense,  of  that  thing  directly  varies  ; 
and,  if  we  have  measured  also  the  general  exchange-value  of 
money  by  means  of  an  average  of  prices,  then,  as  the  average 
price  of  all  things  varies,  so  does  the  cost-value  of  all  things  on 
the  average  directly  vary.  Or  if  we  should  find  that  the  ex- 
change-value of  money  in  all  other  things  has  been  constant, 
but  that  its  cost-value  has  fallen,  then  we  should  know  that  a 
steady  price  of  anything  means  a  fall  in  its  cost-value  to  just  that 
extent,  a  fall  of  price  a  still  greater  fall  in  cost-value,  or  a  rise 
of  price  either  a  lesser  fall  in  cost-value  or  a  stationary  cost- 
value  or  a  rise  in  cost-value,  according  to  the  proportions  between 
the  fall  of  money  in  cost-value  and  the  rise  of  the  price  ;  and 
we  should  know  that  all  other  things  have  on  the  average  fallen 
in  cost-value  to  the  same  extent  as  money.     Or  we  might  find 

the  ton  of  coal  is  one  of  the  costs  in  producing  the  ton  of  iron.  Labor  is  another 
one  of  the  costs  of  producing  the  ton  of  iron  ;  and  as  labor  was  expended  in  pro- 
ducing the  ton  of  coal,  labor  is  an  ultimate  cost  of  production  (and  cost-value  is 
measured  by  labor).  Labor  cannot  be  the  price  of  anything,  because  one  man 
cannot  give  it  to  another. 


126        SELECTION    OF    PARTICULAK    EXCHANGE-VALUES 

that  the  exchangc-vahie  and  the  cost- value  of  money  liave  both 
changed,  the  one  rising  and  the  other  falling,  or  hotli  together 
rising  or  falling,  only  the  one  more  and  the  other  less.  There 
are,  in  fact,  all  the  five  typical  possibilities  we  have  noticed  in 
another  case  between  the  variations  of  two  independent  quantities. 
But  whichever  of  these  changes  take  place,  when  we  know  their 
proportions,  it  is  easy  to  calculate  out  all  the  particular  varia- 
tions concerning  which  knowledge  is  desired.  Therefore  the 
measurement  of  the  cost-value  of  money  is  also  desirable. 

§  5.  Now  the  cost-value  of  anything,  in  the  wide  sense,  as 
determined  by  the  labor  cost  of  acquisition,  and  identical  in 
magnitude  with  esteem-value,  is  measured,  not  by  the  average 
labor-cost  of  producing  the  thing  in  question,  but  by  the  average 
labor-cost  of  procuring  it.  In  the  case  of  paper  money,  it  is 
evident  that  we  have  no  interest  in  the  cost  of  producing  it ; 
and  even  in  the  case  of  metallic  money,  our  interest  in- the  cost 
of  producing  it  is  small,  as  regards  the  subject  before  us.  AVe 
are  interested  in  the  labor-cost  of  procuring  it.  Evidently  such 
a  thing  as  the  general  lal^or-cost  of  procuring  money  is  to  be 
measured  by  the  average  labor-cost  to  the  average  man  of  pro- 
curing money.  This  may  be  found  by  finding  the  total  money- 
earnings  of  a  country  in  a  given  time,  say  an  hour,  and  dividing 
it  by  the  number  of  workers,  with  allowance  for  the  number  of 
hours  a  day  each  one  works.^ 

Here  incidentally  it  deserves  of  notice  that  in  measuring  the  cost 
of  procuring  money,  we  should  not  confine  our  calculation  to  the 
consideration  of  wages,  or  of  wages  and  salaries.  To  confine  at- 
tention to  wages,  and  especially  to  the  wages  of  the  cheapest  and 

2  There  was  a  dispute  in  the  early  part  of  the  nineteeiitli  century  as  to  wliether 
the  measure  of  "  value  "  is  the  quantity  of  labor  needed  to  produce  the  thing  or 
whether  it  is  the  quantity  of  labor  the  thing  will  coturuand  in  exchange.  Adam 
Smith  had  advanced  both  positions,  and  sides  were  taken  by  his  followers, 
liicardo  assuming  the  former  tenet,  while  ^lalthus  defended  the  latter,  each 
having  partisans.  Now  it  is  plain  that  the  measure  of  cod-value  proper  is  the 
average  quantity  of  labor  needed  to  produce  the  thing;  and  it  would  seem  as  if 
the  measure  of  cost-value  in  the  wide  sense,  or  esteem-value,  may  be  taken  to  be 
the  average  quantity  of  labor  the  thing  will  command.  Thus  both  those  positions 
were  correct  and  incorrect.  They  were  correct  each  of  one  kind  of  "value,"  and 
incorrect  of  the  other  kinds  of  "  value."  They  were  both  incorrect  of  exchange- 
value. 


WAGES    EXCLUDED  127 

commonest  sort  of  laborers,  or  to  agricultural  laborers,  as  re('(»ni- 
mendcd  by  some  writers,^  would  be  like  conliuiiig  our  attention 
in  the  attempt  to  measure  the  exchange-value  of  money  to  the 
price  of  wheat  or  other  food,  as  performed  by  some  of  the  earlier 
Avorkers  in  this  field.  Not  only  all  w^ages  and  all  salaries,  but 
all  profits  (a})art  from  rent  and  income,  if  these  be  already 
comited  in  the  gross  income  of  those  who  pay  them),  should  be 
included  in  the  examination,  and  some  means  of  averaging  them 
should  be  found.  The  common  word  for  wages,  salaries,  and 
profits,  is  earnings.  The  general  money-earnings  of  the  hour's 
work  of  all  the  working  part  of  the  community  should  be  some- 
how found.  Perhaps  some  earnings,  as  immeasurable  in  prac- 
tice, would  have  to  be  omitted,  just  as  in  measuring  the  ex- 
change-value of  money  we  have  to  omit  the  prices  of  some 
things.  But  the  eifort  should  be  made  to  include  as  many  as 
possible. 

§  6.  Over  against  this  measurement  of  the  cost- value  of 
money,  we  have  curiosity  to  know  the  cost-value  in  the  same 
sense,  or  esteem-value,  of  all  commodities  in  general.  We 
might  be  tempted  to  measure  this  in  the  same  way,  by  finding 
the  total  product  of  a  country  in  a  given  time  and  dividing  by 
the  number  of  producers,  or  of  consumers.  But  to  this  there  is 
objection,  in  that  the  total  prijduct  would  have  to  be  measured 
all  by  weight  or  all  by  capacity,  and  when  a  change  takes  place 
between  two  periods  a  different  result  would  be  reached  accord- 
ing as  the  one  or  the  other  of  these  measures  were  used — a  diffi- 
culty we  have  already  noticed  in  another  connection.  For  this 
lumping  together  of  a  total  mass  product  would  mix  up  mate- 
rials of  different  qualities  and  fineness,  and  from  period  to  period 
a  change  in  the  mere  total  mass  would  disregard  possible  im- 
provements or  retrogressions  in  the  qualities^ — a  difficulty  we 
shall  meet  again.     Instead,  we  should  have  to  measure  the  vari- 

^  Harris,  Essay  upon  money  and  coins,  1757,  Part  I.,  p.  13  ;  Malthus,  op.  cit., 
pp.  Q(i,  112,  116  ;  Shadwell,  op.  cit.,  pp.  202-203,  and  in  the  Journal  of  tlie  British 
Association  for  the  Advancement  of  Science,  1883,  p.  626 ;  T.  I.  Pollard,  Gold 
and  silver  weighed  in  the  balance :  a  measure  of  their  value,  Calcutta  1886,  p.  75. 

*  What  we  want  really  to  find  Is  the  total  quantity  of  pleasure  procurable 
from  our  productive  activity.  But,  in  equal  quantities,  coarse  goods  do  not 
yield  so  much  pleasure  as  fine  goods. 


128        SELECTIOX    OF    PARTICULAR    EXCHANGE-VALUES 

atioii  ill  the  cost  of  procluction  of  every  kind  of  article  separately, 
by  fiutling  its  total  product  in  a  given  time  and  dividing  by  the 
number  of  its  producers,  at  the  two  periods  comjjared  ;  and  then 
somehow,  by  some  method  of  averaging  never  yet  investigated, 
combine  the  variations  in  the  costs  of  production  of  the  diiferent 
kinds  in  one  variation  (or  constancy)  of  the  cost  of  production 
of  all  things.  This  is  a  very  complex  method.  It  is  the  only 
one  possible  for  measuring  general  cost-value  in  the  narrow  and 
proper  sense.  But^for  the  wider  sense,  in  which  cost- value  be- 
comes almost  identical  with  esteem-value,  another  simpler 
method  is  also  possible. 

This  is  to  find  the  esteem-value  of  money  alone,  and  to  find 
the  exchange-value  of  money  alone  ;  for  by  comparing  the  results 
of  these  two  measurements  we  can  get  the  result  desired.  We 
thus  see  that  we  have  need  of  two  distinct  operations,  which  are 
supplementary  to  each  other,  and  together  help  us  to  reach 
a  final  result  in  regard  to  the  esteem-value  of  commodities  in 
general. 

§  7.  Now  if,  instead  of  these  two  separate  operations,  we 
should  attem})t  to  perform  a  single  operation  by  including  earn- 
ings in  the  same  measurement  with  the  prices  of  commodities, 
we  should  form  a  hodge-podge  that  has  no  meaning.  Its  results 
would  indicate  neither  the  exchange- value  nor  the  esteem-value 
of  money,  and  as  it  is  undertaken  only  with  a  view  to  measur- 
ing the  "  value  "  of  money,  it  would  mean  nothing,  there  being 
no  economic  value  apart  from  the  four  kinds  Ave  have  analyzed 
out,  nor  any  value  compounded  of  any  two  or  more  of  these. 
For  instance,  take  the  case  which  is  believed  to  have  been  going 
on  for  the  past  twenty  odd  years.  Prices  in  general  have  been 
falling,  and  at  the  same  time  money  earnings  in  general  have 
been  rising.  A  whole  school  of  modern  writers,  reviving  views 
which  triiimplied  during  a  similar  jieriod  in  the  early  part  of 
the  century,  have  concluded  from  this  state  of  things  that  the 
"value"  of   money  has   been    about  stationary.^     This    result 

■'■'  E.  (J.,  "  Gold  prices  fell  only  19  per  cent,  from  1873  to  1891.  .  .  .  Wages,  in 
gold,  rose  more  than  14  per  cent,  from  1873  to  1891.  .  .  .  The  advance  in  wages 
since  1873  so  nearly  offsets  the  decline  in  prices  tiiat  when  fairly  tested  by  both 
prices  and  wages  the  value  of  gold  in  1873  and  1891  was  practically  the  same,"  B. 


WACJKS    EXCLUDED  129 

would  be  obtained,  in  fact,  if  we  ])ut  money  earnings,  which 
have  risen,  and  the  prices  of  commodities,  which  have  fallen, 
in  the  same  table,  and,  attaching-  equal  im])<>rtance  or  weight 
to  each,  drew  an  average  between  them  all — on  tlie  su})position 
that  the  rise  of  the  former  lias  been  about  equal  to  the  fall  of 
the  latter.  And  yet  it  is  plain  that  the  fall  of  the  prices  of 
commodities  means  a  rise^of  the  exchange- value  of  money  in 
commodities  ;  and  the  rise  of  people's  earnings  in  money  does 
not  mean  a  fall  of  the  exchange-value  of  money  in  labor,  there 
beini>:  no  such  exchange- value,  monev  not  beino-  cxchano-ed  for 
labor  ;  but  it  means  a  fall  in  the  cost-value  or  esteem-value  of 
money.  Then  .what  "  value  "  is  it  of  money  that  has  remaincnl 
stationary  ?  Surely  there  is  no  value  that  is  a  mean  between 
exchange- value  and  cost-value  (or  esteem-value). 

Again,  if  the  2)rices  of  commodities  should  for  a  time  remain 
constant  on  an  average,  and  if  also  the  average  money-earnings 
of  workers  remain  stationary,  the  same  result  would  be  obtained 
from  the  mixed  method  of  measuring  the  "  value  "  of  money — 
namely  that  tlie  "  value  "  of  money  has  remained  unchanged. 
Yet,  in  this  case,  both  the  exchange-value  and  the  cost-value  or 
esteem-value  of  money  have  remained  unchanged.  Thus  the 
mixed  method  will  give  the  same  result  in  regard  to  the  "  value  " 
of  money,  even  though  the  separate  measurements  of  the  ex- 
change-value and  of  the  cost-value  or  esteem-value  of  money 
give  diiferent  results.  The  separate  methods,  however,  give  us 
another  result  not  indicated  by  the  mixed  method.  In  the  last 
suppositional  case  it  is  evident  that  the  cost-value  and  esteem- 
value  of  commodities  in  general  have  remained  stationary.  But 
in  the  first  actual  exam])le  it  is  plain,  from  the  separate  measure- 
ments, that  the  cost-value  and  esteem-value  of  commodities  in 
general  have  fallen  even  more  than  the  cost-value  and  est(>em- 
value  of  money.  For  a  fall  of  the  esteem-value  of  money  with 
a  rise  of  its  exchange-value,  means  that  men  are  obtaining 
money  more  easily  and  that  their  money  is  purchasing  for 
them  more  commodities,  that  therefore  conunodities  are  being 

W.  Holt,  Interest  and  (tppreciotion,  i^ound  Ciirreiu-y  (Reform  t'lub),  Now  York, 
Nov.  1.'),  1898,  p.  368. 


130        SELECTION    OF    PARTICUI.AR    EXCHANGE-VALUES 

obtained  still  more  easily  than  money,  and  hence  their  cost-value 
and  esteem- value  have  been  falling  still  more  rapidly  than  those 
values  of  money.  The  influence  of  falling  prices  and  of  falling 
esteem-value  of  money  is  cumulative  upon  the  esteem-value  of 
cormnodities — in  our  estimates  of  those  things.  But  the  mixed 
method,  which  indicates  merely  that  an  anomalous  "  value  "  of 
money  is  stationary  as  well  in  the  onea^case  as  in  the  other,  does 
not  distinguish,  does  not  inform  us,  whether  commodities  are 
falling  in  cost-value,  or  esteem-value,  or  whether  they  are  sta- 
tionary. And  still  another  exactly  opposite  example  is  conceiv- 
able, in  which  money-earnings  might  fall  and  prices  rise,  in  which 
therefore  the  cost- value  and  esteem- value  of  money  are  rising  and 
the  exchange-value  of  money  is  falling,  where  the  mixed  method 
of  measuring  might  also  indicate  constancy  in  the  "  value  "  of 
money.  Thus  on  three  very  diiferent  occasions  the  mixed  method 
would  give  the  same  answer,  although  on  two  of  these  occasions 
the  exchange-value  and  the  cost-value  (or  esteem- value)  of  money 
are  acting  in  diametrically  opposite  ways,  and  in  the  other  they 
are  standing  still  in  the  mean.  Also  this  method  might  give 
the  same  answer  no  matter  how  much  the  exchange-value  of 
money  is  falling,  proyided  the  esteem-value  of  money  is  rising 
sufficiently  to  counter-balance,  or  ince  versa.  The  utter  worth- 
lessness  of  such  a  method,  which  mixes  up  distinct  things,  is 
apparent. 

If  such  a  mixed  method  is  worthless  when  we  carry  out  the 
one-half  of  it  thoroughly,  by  including  in  the  list  all  earnings, 
it  is  a  fortiori  worthless  if  we  execute  this  half  of  it  imperfectly 
by  mcluding  in  the  lists  only  wages,  and  still  more  if  only  some 
Avages. 

§  8.  There  has  really  been  gross  confusion  of  thought  in  the 
recommendation  of  this  method — an  extension  to  a  whole  of 
what  belongs  only  to  a  part.  (V)nsidering  only  the  case  of  per- 
sons who  haye  fixed  incomes  and  Avho  spend  some  of  it  in  buy- 
ing commodities  and  some  of  it  in  hiring  servants,  economists 
have  seen  that  if  the  prices  of  commodities  fall  and  the  wages  of 
servants  rise,  there  is  some  tendency  here  toward  com]iensatiou 
and  balancing.      In  attempting  to  measure  the  purchasing  power 


WAGES    EXCI.rDED  131 

of  these  incomes,  therefore,  it  would  be  ]iroper  to  take  account 
of  the  wages  of  servants  along  witli  tiic  ])rices  of  goods.  But 
then  the  prices  of  goods  to  be  used,  in  order  to  observe  parallel- 
ism, shovdd  be  retail  ])riees.  And  investigation  should  be  made 
as  to  how  much,  on  the  average,  is  sj)ent  on  goods  and  ho\v 
much  in  wages,  in  order  to  weight  goods  and  wages  accordingly 
— probably  with  the  result  of  finding  the  weight  of  wages  to  be 
relatively  small,  wherefore,  for  the  counterbalancing,  the  rise  of 
wages  would  have  to  be  considerably  greater  than  the  fall  of 
prices.  In  doing  this,  however,  all  that  is  accomplished  is  the 
construction  of  a  s])ecial  standard  for  a  ])art  of  the  community. 
Then,  forgettmg  the  limitation  of  this  procedure,  and  Avanting  to 
use  wholesale  prices,  and  to  form  a  standard  for  the  whole  commu- 
nity of  producers  (including  the  servants)  and  consumers,  and 
therefore  to  weight  labor  as  equally  important  with  the  products 
of  labor,  some  economists  have  assumed  that  a  compensation 
which  exists  for  a  small  part  of  the  conmiunity  exists  for  the 
community  at  large.  The  compensation  exists,  in  full  plenitude, 
only  in  the  case  of  a  few  persons  who  are,  so  to  speak,  con- 
sumers both  of  commodities  and  of  services,  and  are  not  them- 
selves producers  or  service-renderers.  In  the  case  of  the  per- 
sons who  render  the  services,  or  who  ])roduce  goods,  without 
themselves  hiring  other  servants,  instead  of  com])ensation,  there 
is  cmnulation,  for  they  earn  more  money  and  their  money  pur- 
chases more  goods.  While  in  the  case  of  other  producers  and 
service-renderers  who  are  also  employers,  the  cumulative  influ- 
ence generally  exceeds  the  compensatory.  Moreover,  the  wages 
of  laborers  hired  for  producing  goods  for  sale,  as  before  re- 
marked, are  already  included  in  the  prices  of  the  products. 
Now  if  these  wages  rise  while  these  prices  fall,  either  this  change 
is  at  the  expense  of  the  emjiloyers,  causing  their  profits  to  fall, 
and  so  not  affecting  earnings  in  general,  or  both  the  rise  and  the 
fall  are  compensated  to  the  employers  by  cheapened  methods  of 
production  and  imj)roved  machinery — that  is,  there  is  compen- 
sation of  an  entirely  different  sort. 

§  8.  Of  course  in  making  the  final  mensurement  of  the  esteem- 
value  of  commodities   in  general,  we  nuist   count    earnings  as 


132        SELECTION    OF    rAETICULAR    P:XCHAXGE-VALUES 

equally  iinportant  with  the  prices  of  commodities.  But  this  is 
because  earnings  wholly  belong  in  the  measurement  of  the  cost- 
value  or  esteem-value  of  money,  and  prices  wholly  belong  in 
the  measurement  of  the  exchange-value  of  money.  Then,  when 
a  single  result  is  obtained  from  the  two  measurements,  equal 
importance  is  to  be  attached  to  each. 

But  in  the  mixed  method  it  is  impossible  to  state  what  ought 
to  be  the  weighting  of  commodities  and  of  earnings  as  wholes 
relatively  to  each  other.  This  problem  has  simply  been  ignored 
by  most  of  the  advocates  of  the  mixed  method.  A  couple  of  tlie 
early  ones  **  weighted  wages  at  about  one  third  of  commodity- 
prices,  without  telling  us  why,  but  apparently  having  in  mind 
some  estimate  of  the  relative  amount  of  money  paid  in  wages  and 
in  other  expenses  by  the  then  dominant  class  of  landlords,  or  by 
their  farmers."  Recent  writers  have  been  less  definite.  Some 
have  implied  that  even  wages,  let  alone  earnings,  should  be 
treated  as  equally  important  with  the  prices  of  all  commodities  ; 
while  others  have  even  asserted  that  greater  importance  should 
be  assigned  to  wages  than  to  prices.  Of  course  these  writers  do 
not  mean  that  peo})le  in  general  spend  as  much  or  more  money 
in  paying  servants  than  in  purchasing  goods.  What  they  have 
in  mind  is  an  indistin(!t  notion  that  what  they  want  to  measure 
is  rather  the  cost-value  or  esteem-value  of  money.  Then,  of 
course,  they  should  give  greater  Aveight  to  Avages  than  to  })rices, 
for  the  simple  reason  that  prices  should  not  be  counted  in  that 
measurement  at  all.  Prices  no  more  belong  in  the' measurement 
of  the  cost-value  (or  esteem-value)  of  money  than  do  wages  in 
the  measurement  of  the  cxchang(>- value  of  money. ^  No  wonder, 
then,  that  the  jjersons  wlio  try  to  combine  the  two  distinct 
measurements  of  the  two  distinct  values  are  at  a  loss  as  to  the 
amounts  of  importance  they  shoidd  attach  to  prices  and  to  wages, 

•^  Evelyn  aiid  Young. 

'Sonietinio  before,  Cantillon  said  it  was  a  common  opinion  in  England  that 
of  a  farmer's  income  a  third  goes  to  liis  landlord,  a  third  to  himself,  and  a  third  to 
his  laborers,  Essai  sur  le  commerce,  about  17;52,  Harvard  ed.,  pp.  159-1(50. 

*  Among  the  writers  above  cited  as  mixing  up  these  things,  Giffen  at  times  in- 
clines to  recommend  treatiiig  of  wages  and  salaries  in  a  separate  table,  B.  55,  p. 
131.  But  he  does  not  recognize  this  as  forming  part  in  the  measurement  of  some- 
thing else  thiiii  ('xcliange-valiic. 


WA(;i:s  i;\('i.ii)i:i)  133 

and  waver  between  them  aeeordinii;  as  the  idea  of  cxchanoe-valiie 
or  the  idea  of  eost-value  predominates  in  tlieir  minds. 

^  10.  Slightly  different,  tliouii'h  at  the  hottoni  similar,  is  the 
position  held  by  a  few  eeonomists  and  many  ])o]itieians  to-day 
that  wao-es  are  a  better  measnre,  or  criterion,  of  the  "value"  of 
money  than  the  prices  of  commodities.  Here  the  opinion  seems 
to  be  that  we  may  make  one  measunnnent  of  the  "value"  of 
money  by  means  of  prices  and  another  by  means  of  wa^es,  with 
dilferent  rcsnlts,  and  that  of  the  two  the  measni-ement  ])y  wages 
is  the  better  or  more  trustworthy.  This  position  is  naive.  The 
measurement  of  the  value  of  monev  bv  waa-es  alone  is  wrono-,  in 
the  first  place,  because  wages  are  not  the  only  earnings.  And, 
in  the  second  place,  a  correctly  performed  measurement  by  earn- 
ings would  be  th(>  measurement  of  the  esteem-value  of  money  ; 
wherefore,  as  such,  it  cannot  be  comj)ared  with  the  measurement 
by  prices,  which  is  not  a  measurement  of  the  esteem-value  of 
money  at  all.  And  the  measurement  by  prices  is  the  measure- 
ment of  the  exchange-value  of  money  ;  wherefore,  as  such,  it 
cannot  be  comjiared  with  the  measurement  by  earnings,  which 
is  not  a  measurement  of  the  exchange-value  of  money  at  all. 
Each  of  these  methods  of  measuring  the  "value"  of  money,  if 
carried  out  in  the  best  manner  possible,  is  the  best  in  its  kind. 
And  neither  has  a  better  claim  than  the  other  to  be  the  better 
measurement  of  the  "  value  "  of  money  singly,  since  the  one 
kind  of  value  is  as  much  value  as  tlie  other.'^ 

^  It  is  hardly  necessary  to  notice  that  sometimes  tlie  same  economists  who 
make  the  above  comparison,  and  sometimes  others,  also  bring  in  still  another 
measurement  of  the  "value  of  money  "  as  equally  good  as,  oreven  as  better  than, 
either  of  the  preceding.  This  is  the  measurement  of  something  which  is  not  value 
at  all  in  any  of  the  economic  senses  of  this  term,  but  to  which  this  term  is  applied, 
in  connection  with  a  similarly  uneconomic  meaning  of  the  term  "  money,"  in  the 
slang  of  the  ' '  street. "  For  certain  classes  of  business  men  often  mean  by  ' '  money" 
loanable  capital,  and  by  the  "value"  of  this  "money"  the  rate  of  discount. 
Naturally  the  measurement  of  this  "value  of  money  "  is  by  the  rate  of  discount 
itself;  and  such  a  measurement  is  a  perfectly  good  one  of  the  "  value  of  money  " 
in  these  meanings  of  the  terms.  But  it  is  not  at  all  a  measurement  of  tlie  "  value  " 
of  money  in  any  of  its  economic  senses;  and  so  cannot  properly  be  In-ought  into 
comparison  with  any  of  tlie  methods  of  measuring  any  of  the  economic  values  of 
money. — There  is  even  another  uneconomic  meaning  of  "value,"  as  used  in  tlie 
old  monetary  science,  namely  the  meaning  of  "intrinsic  value"  in  the  case  of 
metallic  money,  this  being  tlie  fine  metal  in  the  coins.  The  measurement  of  this 
"  value  "  is  the  simplest  thing  in  tlic  world,  l)eiiig  made  by  finding  the  weight  of 


134        SELECTION    OF     PARTICULAR    EXCHANGE- VALUES 

§  11.  Of  course  if  either  of  these  methods  is  thrust  forward 
into  the  province  of  the  other,  it  should  be  driven  lx\ck.  But 
what  happens  mostly  is  that  one  person  in  speaking  of  "  value  " 
is  proue  to  think  oidy  of  the  one  kind  of  value,  and  then  he 
claims  that  the  measurement  properly  applicable  to  that  kind  of 
value  is  the  better  measurement  of  "  value."  He  ought  really 
to  say  that  it  is  the  only  measurement  of  "  value,"  and  dismiss 
the  other  altogether,  confining  himself  to  the  one  meaning  of  the 
term  he  uses  ;  only  this  he  cannot  do,  liecause  the  other  niean- 
iuff  of  "value"  ahvavs  does  lurk  in  his  mind.  The  best  and 
only  thing  to  do  is  to  distinguish  what  kind  of  value  we  are 
treating  of,  and  to  assign  to  each  kind  its  own  pr()])er  kind  of 
measurement.  Otherwise  when  two  persons  meet  each  other  in 
the  combat  of  opinions,  who  happen  each  to  be  thinking  of  a 
different  kind  of  value  although  they  are  both  talking  simply  of 
"value,"  they  will  be  a])t  to  fall  into  behavior  very  much  like 
that  of  the  two  knights  on  the  opposite  sides  of  the  shield. 

And  yet,  to  repeat,  unlike  those  two  knights,  each  of  these 
contending  parties  has  an  inkling  of  the  other's  meaning,  and 
adopts  it  himself  at  times.  Here  is  the  fundamental  explanation 
whv  tmv  economists  have  advised  such  a  mixing  of  incompatible 
elements  in  the  measurement  of  the  "  value  "  of  money.  Using 
simply  the  one  term  "  value,"  they  have  not  distinguished  which 
kind  of  value  they  were  seeking  to  measure,  and  so  have  allowed 
elements  ]n-oper  in  the  measurement  of  distinct  kinds  to  creep 
into  a  single  measurement,  which  then  is  really  a  measurement 
of  no  one  kind  of  value.  Yet,  as  we  have  already  had  occasion 
to  remark,  many  of  the  most  prominent  economists  have  af- 
firmed that  when  they  used  the  term  "  value  "  they  intended  to 
refer  only  to  "  exchangeable  value."  .Vnd  in  truth  most  of  the 
eifbrts  of  economists  have  hitherto  been  directed  to  measure 
princi]ially  the  exchange-value  of  moniy,  with  a  little  infiltra- 
tion of  cost- value  elements.  But  cost-value  itself,  freed  from 
any  other  kind  of  value,  no  one  has  ever  yet  attem])ted  to  meas- 
ure, or  so  nuich    as  mentioned  as  a  desirable  object  of  mensu- 

the  coins  and  allowing  for  the  alloy.  It  would  be  no  more  aljsurd  to  bring  this 
into  the  comparison  and  to  say  that  the  best  measurement  of  the  "value"  of 
money  is  Ijy  the  weiglit  and  lineness  of  the  coins. 


WAGES    EXCLUDED  135 

ration  by  itself.'"  Thus  tlio  mensuration  of  exchange-value  has 
been  brought  well  along,  while  the  measurement  of  cost-value 
has  hardly  so  much  as  been  begun.  There  have  been,  and  es- 
pecially nowadays  are,  a  ntlinber  of  economists  who  look  upon 
cost-value  as  a  more  important  kind  of  value  than  exchange- 
value, — who,  for  instance,  maintain  that  money,  the  recognized 
measure  of  "  value,"  should  be  a  fixed  standard  rather  of  cost- 
value  than  of  exchange-value.  Upon  these,  then,  the  duty  is 
incumbent  of  instituting  the  attempt  to  measure  the  constancy 
or  variation  of  the  cost-value,  or  esteem-value,  of  mf)ney. 

The  present  work,  to  repeat,  is  concerned  only  with  the  men- 
suration of  exchange-value. 

^^  To  be  sure,  Ricardo  and  Malthus  and  others,  we  have  seen,  were  mostly  in- 
terested in  the  measurement  of  cost-value  or  of  esteem-value.  But  they  always 
expounded  their  procedure  as  a  search  after  the  "  real  "  measure  of  "  value  in  ex- 
change," so  that  the  measurement  of  exchange-vaUie  always  interfered  with  their 
measurements  of  the  other  kinds  of  value. 


CHAPTER   Y. 

>rATIIEMATrCAL    FORMULATION  OF    EXCHANGE- VALUE 
RELATIONS. 

I. 

§  1 .  For  working-  out  our  ])rol)leni  of  measuring  the  exchange- 
value  of  anything  by  combining  its  particukir  exchange-values 
into  its  one  general  exchange-value,  or  the  many  variations  of 
these  into  one  variation,  the  aid  of  mathematical  formulation  is 
indis})ensable. 

The  money-unit,  being  our  unit  for  measuring  particular  con- 
temporaneous ex  change- values  at  the  same  place,  has  become  also 
our  ])ractical  small  unit  for  measuring  the  comparative  exchange- 
values  of  things  in  dift'erent  places  and  their  constancy  or  varia- 
tion at  different  times.  But  it  is  a  variable  unit,  comparable 
with  metallic  or  wooden  foot-sticks,  which  vary  in  length  at  dif- 
ferent temperatures  and  which  also  wear  down,  except  that  the 
money-unit  may  vary  in  exchange-value  indefinitely  in  either 
direction,  and  so  is  a  much  less  trustworthy  practical  unit.  To 
be  precise,  it  is  not  the  money-unit  itself,  not  the  coin-unit — the 
material  thing, — that  is  the  unit  of  exchange-value,  as  also  not 
the  material  stick  is  the  unit  of  length  ;  but  as  the  length  of  a 
particular  stick  is  the  unit  of  Icugtli,  so  the  exchange-value  of 
the  money-unit  is  the  unit  of  exchange- value. ^  Now  just  as  the 
surveyor  corrects  the  variations  of  his  measuring  instruments  in 
their  attribute  of  length,  so  we  want  to  be  able  to  correct  the 
variations  of  our  economic  uKnisuriiig  instruments  in  their  attri- 
bute of  exchange-value,  and  wish  to  construct  a  true  and  unvary- 
ing unit  underlying  our  practical  units,  the  same  in  all  })laces 

1  That,  strictly  speaking,  we  measure  the  value  of  things  not  by  money,  but 
by  value  of  money,  see  Prince-Smith,  op.  cit.,  p.  .')()9 ;  Knies,  Das  Geld,  2tl  ed., 
1885,  p.  150;  cf.  Rossi,  op.  cit.,  pp.  145,  157. 

136 


OF  ExrirAX(ii:-vAi-ri-;  ijei.atioxs  l;J7 

and  at  all  times.  Some  of  the  older  writers  on  monetary  mat- 
ters asserted  that  because  mt^ney  is  the  measure  of  other  things, 
we  cannot  measure  it."  This  is  a  false  position  in  metrology, 
unhappily  not  yet  wholly  ahandone*!.'^  All  our  practical  meas- 
ures must  themselves  be  measured — and  not  only  our  ordinary 
measures,  but  also  the  so-called  standard  measures.^  In  fact, 
metrology  itself  is  the  science  of  measuring  our  measures.  Of 
course,  the  measurement  of  our  measures  is  different  from  tlie 
measurement  of  other  thiup-s.  We  measure  other  things  bv  com- 
paring  them  with  our  measures — a  simple  operation.  We  meas- 
ure our  measures  l)y  comparing  them  with  other  things,  prefer- 
ably with  some  selected  other  things — a  complex  operation. 
The  ol)ject  of  measuring  other  things  is  to  find  their  relative 
sizes,  as  well  as  the  constancy  or  variation  of  their  sizes,  by  find- 
ing their  sizes  compared  with  one  given  and  constant  size,  the 
measure  used.  The  object  of  measuring  our  measures  is  to  find, 
except  in  the  case  of  different  units  of  the  same  measure,  only 
their  equalitv'  or  inequalities  and  their  constancy  or  variations. 
This  is  the  labor  we  have  before  us  in  regard  to  exchange-value. 
We  want  especially  to  measure  our  measure  of  exchange-value, 
money. 

As  being  the  simpler  form  of  our  problem,  we  may  devote 
most  of  the  enquiry  to  the  attempt  to  measure  the  constancy  or 
variation   of  the   money-unit    in   exchange-value    at  one  place 

2  Javolinus  in  Digest,  XLVI.,  I.,  42  ;  followed  l)y  Molinaeus  and  Budelius  (iu 
Thesaurus's  collection  of  monetary  tracts,  Turin  1609,  pp.  236,  461). — The  doc- 
trine was  an  inference  from  Aristotle's  statement  that  money  measures  all  things 
{Bth.  N.  V,  V,  10).  Hence  Turgot  also  said  that  money  can  be  measured  only  by 
other  money,  op.  cit.,  p.  76. 

'L.  A.  Garnett,  The  crux  of  the  money  question:  has  gold  risen f  Forum, 
New  York,  Jan.  1895,  p.  581. — And  Mannequin  has  gone  so  far  as  to  say  not  only 
that  we  do  not  measure  our  measures,  but  that  it  is  not  needed  that  measures  be 
stable,  this  being  a  prejudice,  and  even,  as  measures  should  be  like  the  measured, 
they  ought  to  be  variable  !  op.  cit.,  pp.  28,  70,  Unijiormite  monetaire,  18G7,  pp.  10- 
11  n.,  La  monnaie  et  le  double  etalon,  1874,  p.  39  (quoted  by  Walker,  Money,  p. 

281  «). 

*  That  money  is  measured  bj'  other  things  was  said  by  Montanari,  Delia  nioneta, 
1683,  ed.  Custodi,  p.  91;  Galiani,  op.  cit..  Vol.  I.,  p.  52;  Condillac,  Le  commerce 
et  le  gouvernement,  1776,  ed.,  1821,  p.  93;  (cf.  Adam  Smith,  op.  cit.,  p.  190)  ; 
Kicardo,  p.  293  ;  Levasseur,  B.  18,  pp.  137-138. — McCulloch  concluded  that 
money  is  therefore  not  a  measure  of  value,  as  it  no  more  measures  other  things 
than  it  is  measured  by  them,  Treatises  and  essays,  18.59,  p.  10. 


]:>.S  MATHEMATICAL    FORMII.ATK  )N 

tlinmgh  tlu'  course  of  time.  The  consideration  of  two  periods 
will  generally  be  sufficient.  And  for  couveuience  we  may  sup- 
})0.se,  at  the  commencement,  that  all  the  classes  of  things  are  of 
equal  exchange- value  importance,  so  that  we  may  employ  even 
^veightiug,  which  is  much  the  simplest  to  deal  with  first. 

§  "2.  Let  us  represent  the  money-unit  by  the  capital  letter  M, 
and  the  habitually  used  mass-units  of  various  classes  of  things 
— say  a  bushel  of  wheat,  a  pound  of  sugar,  a  hiuidredweight  of 
iron,  a  ton  of  coal — by  tiie  (•a})ital  letters  A,  B,  C,  and  so  on. 
Or  if  the  reader  prefers,  he  may  understand  by  these  letters  al- 
ways the  same  mass-unit — say,  always  a  pound  of  ^\heat,  a 
pound  of  sugar,  a  pound  of  iron,  a  ])ound  of  coal ; — for  we  shall 
■  see  that  in  most  cases  no  difTerence  residts  from  ado])ting  either 
plan.  From  henceforth,  the  ca])ital  letters  being  a})pro])riated 
for  this  purpose,  we  can  no  longer  refer  to  the  classes  themselves 
indefinitely  by  the  plahi  capitals  ;  Init  we  may  do  so  by  putting 
the  capitals  in  brackets,  so  that,  for  instance,  we  shall  say  that  A 
represents  a  mass-unit  of  [A],  a  class  of  things.'^  AVe  may  su])- 
pose  that  at  the  first  or  basic  period  M  will  purchase  a  A,  6  B, 
c  C,  and  so  on — «,  h,  c, indicating  the  numbers  (above,  be- 
low, or  at  unity)  of  the  mass-units  of  the  classes  Avhich  one  M 
will  purchase.  Using  the  sign  c=,  as  before,  to  express  equiva- 
lence, we  may  express  this  state  of  things  as  follows, 

M  r  oA  r  6B  ccC  = 


At  the  second  period  we  may  suppose  that   M   will  purchase 

a'a  A,  h'b  B,  c'c  C,  and  so  on — a'((,  h'b,  c'c, representing  the 

new  numbers  of  the  mass-units  of  each  class  now  purchasable 

with  M  ;  wherefore  «',  h' ,  c' , ,  according  as  they  are  at  unity 

or  as  they  depart  from  unity,  above  or  below,  express  the  con- 
stancy or  variation,  rise  or  fall,  of  the  (piantity  of  each  class 
which  one  M  will  purchase  at  the  second  period  conqxired  with 
the  first,  conse({uently  the  constancy  or  variation,  rise  or  fall,  of 
M's  particular  exchange-values  in  each  class."     It  is  plain  that 

5  This  device  of  notation  is  borrowed,  with  a  slight  change,  from  Walras. 

*  E.  g.,  if  one  dollar  at  first  purchases  2  bushels  of  wheat  ( A,  for  instance,  repre- 
senting 1  bu.  of  wheat)  if  it  later  purchases  2  bushels  of  wheat,  then  a  =  2,  a'a  =  3, 
and  a'  =  5  =  11,  wliich  indicates  a  rise  of  .")()  per  cent,  in  M's  exchange-value  in 


OF    KXCHAXCiK-VAMK    IfKI-ATrOXS  1  .'U) 

the  fiii'iircs  a,  h,  r, will    he   vai'ioiis  according  to  the  nias.s- 

units  whieh  ha])|)eii  to  be  chosen,  l)ut,  the  nias.s-units  once 
chosen  being;  of"  course  used  at  both  periods,  the  variations  repre- 
sented by  a',  b' ,  c', are  the  same,  no   matter  what  be  the 

mass-units  used.      Now  at  the  second  period  we  liave 

M  =  «'a  A  --6'6B  c  c'cC  = 

These  expressions,  however,  are  not  equations.  They  do  not 
express  equality  between  quantities  of  the  same  sort,  but  equiv- 
alence (/.  e.,  equality  in  one  kind  of  quantity,  exchange-value) 
between  quantities  of  another  or  other  kinds  (weight,  capacity, 
etc.,  of  different  things).     Therefore  they  are  not  satisfactory. 

§  -3.  Let  us  proceed  anew  by  representing  the  general   ex- 
change-value of  M,  of  A,  of  B,  of  C,  • ,  by  the  italicized  capital 

letters  M,  A,  B,  C, ,     These  exchange-values  we  shall  need 

to  distinguish  at  the  two  periods,  which  we  may  do  by  ap- 
pending a  small  number  to  them.  When  we  wish  to  distinguish 
between  general  exchange-value  in  all  other  things  and  general 
exchange-value  in  all  things  (including  the  thing  itself),  wt  may 
do  so  by  ap})ending  a  small  o,  or  a  small  a.  Thus  M,,^  means 
the  exchange-value  of  the  money -unit  in  all  other  things  at  the 
first  period.  Therefore  the  expression  M  =c=  aA  may  (by  Prop- 
osition VII.)  be  changed  into  M^  =  aA^,  which  means  either 
that  the  general  exchange-value  of  M  is  equal  to  the  general 
exchange- value  of  a  A,  or  that  the  general  exchange-value  of  M 
is  (I  times  the  general  exchange-value  of  A,  at  the  first  period. 
We  may  then  have 

J/j  =  aA^  =  bB^  =  r(:\  =  , 

and 

31  =  a' a  A.,  =  b'bB.,  =  c'cC= 


Here  we  have  equations  proper  ;  for  the  terms  all  refer  to  equal 
homogeneous  quantities,  namely  exchange-values,  although  these 
are  attributes  possessed  by  different  quantities  of  various  articles 

[A] .  If  the  figures  were  reversed,  we  should  have  a'  =  ^,  indicating  a  fall  of  33J 
percent,  in  M's  exchange-value  in  [A].     If  M  purchases  the  same  quantity  at 

both  periods,  a' a  =  a,  whence  a'  =       =1,  indicating  constancy. 


140  MA  l"lli;.\[A'l"l(Al.     lOnMlLATIOX 

variously  lucasurcd  as  ix'tiards  tlu'ir  masses.  To  say  the  ex- 
change-valiu'  of  an  ounce  of  gold  is  e(|Ual  to  the  cxeliange-value 
of  twenty  bushels  of  wheat  is  to  express  equality  as  much  as  to 
say  the  weight  of  a  bushel  of  wheat  is  e(|ua]  to  the  weight  of  so 
or  so  many  cubic  inches  of  water.  It  is  well  known  that  a 
pound  of  feathers  is  as  heavy  as  a  pound  of  lead." 

These  true  mathematical  equations  admit  of  mathematical 
treatment.      Thus  from  the  equation  J/j  =  aA^  we  may  derive 

^4,  =  —  JI.  ;  and  from  JZ,  =  a' a  A.,,  aA„  =  — ,  J/.,  or  ^4.,  =     ,    J/„ 

— which  mean  that  a  certain  quantity  of  [A] ,  and  consequently 
[A]    in  general,  has  varied  in   exchange-value  **  compared  with 

[Al]  by  — ,  the  reciprocal  of  []M]  's  variijtiou  in  [A]  (cf.  Prop- 
ositions IX.  and  XIIL).  If  wc  knew  that  ^l,  =  Mj,  we  should 
know  that  a'a.[„=  ((A,,  whence  ^4.,=  — ,  ^4,  which  means  that 
A,  or  [.V]  in  general,  has  varied  in  general  exchange- value  by 
—  (in  agreement  with  Proposition  XXXIIL).      Or  if  we  knew 

that  ilig  has  varied  from  J/^  in  a  certain  pro])ortion,  Ave  should 
know  that  a'aA.,  has  varied  from  ((A^  in  the  same  pro])ortion, 
the  general  exchange-value  of  A,  and  of  [A],  thus  undergoing 
a  double  variation  (in  which  the  one  may  enhance,  lessen,  neu- 
tralize, or  outdo  the  other).  As  yet,  however,  we  do  not  know 
the  relation  of  J/,  to  il/j,  this  being  what  Ave  Avant  to  find. 

Still,  for  this  purpose,  the  expressions  are  not  serviceable  in 
the  meanings  so  far  ascribed  to  them.  For  Ave  cannot  combine 
the  general  exchange- values  of  different  things  in  order  to  find 
therefrcmi  the  general  exchanp-e-value  of  auA'  one  thinu'.     AVhat 

"  Bourguin  says  there  arc  no  equations  between  excliange-values,  beeause  ex- 
change-value is  not  intrinsic,  B.  i;'.2,  p.  38.  The  reason  is  not  adequate.  We  have 
seen  that  even  weigiit  is  not  intrinsic.  Certainly  wealth  is  not ;  yet  two  men  may 
be  equally  rich. 

"  If  we  are  dealing  with  M^,  the  general  exchange-value  in  which  [A]  so  varies 
is  likewise  J,,.  But  if  we  are  dealing  with  31^,  the  general  exchange-value  in 
which  [A]  so  varies  is  not  A^,  but  a  general  exchauge-value  of  [A]  in  all  things 
other  to  money,  i.  e.,  in  all  commodities,  including  itself.  Another  distinction 
of  a  similar  nature  will  be  noticed  in  ?  4. 


OF    i:XCHAN(;K-VAI.rK    I.'ELATION.S  141 

we  can  so  combine  are  the  ])articular  exchange-values  of  the 
thmg  in  (piestion  in  the  other  ckisses  of  things.  Now  by  Prop- 
osition YII.  a  thing's  ])articuUir  exchange-vahie  in  anything  else 
is  equal  to  its  general  exchange-value  ;  and  by  Proposition  VI. 
the  thing's  general  exchange-value  is  equal  to  the  general  ex- 
change-value of  the  quantity  of  the  other  thing  it  exchanges  i'or, 
by  which  quantity  of  that  thmg,  according  to  Proposition  I.,  its 
exchange-value  in  that  kind  of  thing  is  measured.  Therefore  a 
thing's  particular  exchange-value  in  another  thing  is  equal  to  the 
general  exchange-value  of  that  (piantity  of  the  other  thing  it  is 
measured  by.  Hence  the  expression  aA^^  may  re[)resent  not 
only  the  general  exchange-value  of  a  A,  but  also  the  particular 
exchange- value  M  in  [A],  at  the  first  period.  And  so  the 
others.  But  it  is  necessary  that  the  symbols  should  distinguish 
whether  they  refer  to  the  general  exchange- values  of  things,  or 
to  the  particular  exchange- value  of  M  in  those  things.  AYe  may 
make  them  do  so  by  always  placing  them  in  marks  of  paren- 
thesis, thus  {a A),  (bB),  (cC), ,  when  referring  to  the  partic- 
ular exchange- value  of  M  in  [A],  in  [B],  in  [C],  ,  at  the 

first  period.  Then  the  expressions  for  the  particular  exchange- 
values  of   M    in  these    things    at    the   second  period   may  be 

written  either  (a'aA),  {h'bB),  {c'cC), ,  or  a'{aA),  b'[bB), 

g'{gG), The  first  of  these  forms  would  be  merely  the  ex- 
pressions for  the  particular  exchange-values  of  M  in  [A] ,  in  [B] , 
in  [C] ,  ,  as  measured  by  the  quantities  of  these  things  it  ex- 
changes for  at  the  second  period.  But  the  second,  while  mean- 
ing this,  mean,  and  express,  something  mcjre,  namely  that  the 
particular  exchange-value  of  ]\I  in  [A] ,  for  instance,  is  a' 
times  what  it  was  before,  that  is,  at  the  second  ])eriod  a'  times 
the  particular  exchange- value  of  M  in  [A]  at  the  first  period  ; 
wherefore  in  this  form  of  the  expression  obviously — and  also  in 
the  other — the  contained  expression  {aA)  still  refers  to  the  old 
exchange-value  of  M  in  [A];  and  so  with  the  rest.  Hence  it 
is  no  longer  necessary  here  to  numl)er  the  expressions  A,  B, 
C  '■> 

»  If  we  knew  that  Mq,^  =  M^^,  tlien  we  should  know  that  (lA,  if  it  refers  to  the 
general  exelmnge-value  of  a  A,  refers  to  something  different  at  the  second  period 
from  what  the  general  exchange-value  of  a  A  was  at  the  first  period,  that  is,  it 


142  MATHEMATUAlv    F<  )HMrj.ATI()X 

AVe  then  have 

M^  =  {aA)  =  {bB)  =  {cC)= , 

which  may  be  more  specifically  written 

Moi  =  {a A)  =  {bB)  =  (cC)  = to  yi  terms, 

n  representing  the  number  of  all  the  (equally  important)  classes 
of  things  employed.  This  means  that  the  general  exchange- 
value  of  the  money-unit  in  all  other  things  is  equal  to  every  one 
of  its  particular  exchange-values,  which  are  stated  as  they  are 
at  the  first  period.     And  we  have 

Mo.  =  {a'aA)  =  {h'hB)  =  {c'cC)  = to  n  terms, 

which  similarly  expresses  the  equality  between  the  general  ex- 
change-value of  the  money-unit  in  all  other  things  and  every 
one  of  its  particular  exchange-values,  which  are  stated  as  they 
are  at  the  second  period.     But  if  we  write  this  as  follows, 

3Io,  =  a'{aA)  =  (/(bB)  =  c'{eC)  = to  n  terms, 

it  expresses  the  same  equality,  but  states  the  ])articular  exchange- 
values  at  the  second  period  as  they  relate  to  what  they  were  at 
the  first  period, 

§  4.  Using  these  running  equations,  ^^'v  can  now  form  expres- 
sions for  the  combination  of  the  ])articular  exchange-values  of 

refers  to  the  general  exchange-value  of  a  A  at  the  second  period,  wherefore  it  should 
be  distinguished  by  a  number,  viz.,  aAi-  But  in  referring  merely  to  the  particular 
exchange-value  of  M  in  [A]  when  we  do  not  know  what  the  relation  between 
3[q.^  and  31  q-^  is,  but  are  seeking  it,  we  are  using  the  particular  exchange- value  of 
M  in  [A]  at  the  first  period  as  our  standard,  and  must  look  upon  the  particular 
exchange-value  of  ^I  in  [A]  as  changed  at  the  second  period  by  a' ,  and  therefore 
we  must  take  (aA)  as  the  constant  and  a'  [a A)  as  the  variable.  But  (ciA)  alone 
is  not  our  whole  standard  ;  it  is  only  part  of  the  whole  standard  composed  of  all 
other  things.  Yet  we  have  to  treat  each  of  these  separately  as  a  standard,  before 
we  can  unite  them.  Then  after  we  have  united  them,  and  when  we  use  the  whole 
composed  of  them  as  the  true  standard,  it  is  no  contradiction  with  the  preceding 
that  we  may  find  aA  at  the  second  period  to  be  different  from  a  A  at  the  first.  The 
particular  exchange-value  of  M  in  [A]  has  changed  by  a'  (while  the  particular 

exchange-value  of  A  in  [^I]  has  changed  by  -,-) ;   yet,  in  the  case  supposed,  the 

particular  exchange-value  of  M  in  [A],  now  a' (aA),  is  of  the  same  magnitude  as 
was  the  particular  exchange-value  of  .M  in  [A]  when  it  was  {aA),  because  at 
each  period  it  is  ecjual  to  the  genei-al  exchange-value  of  AI,  which  is  supposed  to 
be  unc'hanged.  Tiie  contradiction  is  only  apparent,  l)Ocause  of  the  change  in  the 
standard  used.     Cf.  Chap.  11.,  Sect.  I.,  ?  1. 


OF    P^XC'HANOK-VAIA'E    RELATIONS  143 

M  in  every  other  thing  into  its  cue  gc^neral  exchange-vahic  in 
all  other  things.  We  can,  in  fact,  mathematically  form  several 
such  combinations,  but  may  content  ourselves  with  three.  To 
begin  with  the  first  period.      From 

J/^j  =  («.l)  =  (bB)  =  (eC)= to  n  terms 

we  perceive  that  tlie  sum  total  of  the  exchange-values  of  nM 
will  be 

nJIg^  =  {a A)  +  (bB)  +  (cC)  -(- to  n  terms ; 

whence 

31,,  =  1  {(aA)  +  (bA)  +  (cC)  + to  n  terms}.'«    (1,  i) 

Again,  the  serial  e(juation  may  l)e  converted  into  this, 

1111  ,        . 
= =          = = to  n  terms, 

J/oi       {a  A)      {bB)     (cC) 

(which  means  that  the  exchange- value  of  a  A  in  [M]  is  equal 
to  the  exchange-value  of  b  B  in  [M] ,  and  so  on,  and  to  that  of 
M  in  [M],  in  agreement  with  Proposition  IV.).  By  the  same 
treatment  this  yields 

0,  1  1  1 

_j j :^ — (-  to  n  terms, 


3Io,      (aA)  '  (bB)  '  (cC) 
whence 

1 
^'^01  =  ^^     2             1             1  ^  '  ^  '  '^ 

+  /i~n\  +  /  y7\  + to  ?j  terms 


n  \  (aA)  ^  {bB)  ^  {cC) 

Lastly,  from  the  first  serial  equation,  and  also  from  its  inverted 
form,  the  total  product  of  the  exchange-values  of  M  taken  /* 
times  will  be 

J/j;  =  {a A)  ■  {bB)  -{eC) to  7i  terms, 

whence 


^^01  =  y\aA)  ■  {bB)  ■  {cC) to  n  terms.  (1,  :->) 

'"  With  diflferent  notation,  anil  also  with  reference  to  the  quantities  of  things 
instead  of  their  exchange-vahies,  this  form  of  the  foriuuhi  was,  perhaps  first,  given 
by  Prince-Smith,  op.  cit.,  p.  372.  Prince-Smith  reached  this  formula  in  some- 
what the  same  way  as  it  is  here  reached  ;  but  he  fiiiled  to  see  that  there  are  other 
formulse  equally  well  deducible  from  the  same  original  running  formula.  The 
idea  expressed  by  this  formula  is  also  alone  advanced  by  Nicholson,  B.  94,  pp. 
307-308,  and  by  Fonda,  B.  127,  p.  Kil. 


144  :srATiii;MATi(AT.  foumulatiox 

Xatui-ally  the  serial  equation  for  the  seeond  ])erio(l  may  be 
treated  in  the  same  way,  and,  likewise  with  even  weighting,  in 
its  first  form,  it  will  yield,  with  omission  of  the  su])('rflnons  "to 
n  terms," 

J/,.,  =  1  iia'aA)  +  {b'bB)  +  {c'cC)  +  '  ("i,  .) 

If     — ^ (•>   -A 

31,,  =  ^/(«'a.i)  ■  {fyiJWlc'cC)  ■ ..:... .  (2, .) 

What  is  here  done  for  M,  be  it  here  parenthetically  remarked, 
can  be  done  for  any  of  the  other  classes.  For  example  at  the 
first  period,  if  we  want  to  express  the  general  exchange-value 
of  A  directly  in  all  the  others,  we  may  begin  by  representing 
the  general  exchange-value  of  a  A  by  a  A^j^,  and  then  {31)  would 

represent  the  exchange- value  of  a  A  in  money,  and  (bB),  (cC), 

will  contii\ue  to  express  the  exchange  value  of  a  A  in  [B],  in 
[C], Then  from 

aA,j,  =  (31)  =  {bB)  =  (cC)  = to  n  terms, 

we  can  derive 

aA,,,  =  -r{{M)  +  {bB)  +  (cC)  + to  n  terms}, 

as  in  the  first  way  ;  and  similarly  in  the  other  two  ways,  which 
may  be  omitted.  This  fi)rmula  we  could  have  derived  directly 
from  formula  1,  by  simply  substituting  tt^l^i  foi*  -l^oi  ''"^^  i-^^) 
for  (aA) ;  and  so  on  in  the  other  omitted  ways.  And  from  this 
fornuila  we  derive 

^o^=^l{W  +  {f^-B)-^{cC)  + }. 

Similarly  for  the  second  jieriod  we  can  get 

a'aAo,  =  I  {{31)  +  (b'bB)  +  {c'eC)  + }, 

and 


^^o.  =  J^^{{m  +  (f>'bn)  +  (r'cC)  + \ 


OF    EXCIIANCJK-VAT.rE    ItHl.ATloNS  145 

It  deserves,  however,  to  be  noticed  tliut  if  we  should  use  these 
formulae  to  calculate  the  variation  of  the  jj^encrul  exchange-value 
of  A  in  the  way  to  be  described  presently,  we  should  obtain  a 
result  different  from  that  obtained  by  calculating  first  the  vari- 
ation of  the  general  exchauge- value  of  M  and  then  the  variation 
of  A  by  means  of  its  variation  in  the  varied  money.  For  these 
two  methods  use  different  standards,  the  class  A  itself  forming 
part  of  the  standard  iu  the  latter  method,  but  not  in  the  former, 
and  money  forming  part  of  the  standard  in  the  former,  but  not 
m  the  latter.      (Of  Propositions  XII.  and  XIII.) 

The  interpretation  of  tlie  above  three  kinds  of  formuhe  is  as 
follows.  They  each  express  primarily  a  way  m  which  the  ex- 
change-value in  all  other  things  of  the  money-unit  (or  of  the  mass- 
unit  of  anything  else)  is  to  be  conceived,  the  first,  that  it  is  to  be  con- 
ceived as  equal  to  the  sum,  of  its  excliangc-values  in  every  one  of  the 
other  things,  or  to  the  sum  of  all  its particuJar  exchange-values,  di- 
vided by  theii'  number  ;  the  second,  as  equal  to  the  reciprocal  of  the 
sum  of  the  reciprocals  of  all  its  particular  exchange-values  divided  by 
their  numher  ;  the  third,  as  equal  to  the  n""  root  of  the  product  of 
all  its  pariicidar  exchange-values,  n  in  number. 

Each  of  these  formulae  is  nothing  but  the  expression  of  a 
mathematical  mean  or  average  between  two  or  more  equally  im- 
portant quantities — the  first,  of  the  arithmetic,  the  second,  of  the 
harmonic,  the  third,  of  the  geometric.  And  each  is  perfectly 
true  as  an  expression  of  its  own  kind  of  average  drawn  betw^een 
the  quantities  given.  But  not  more  than  one  of  these  averages 
— and  perhaps  none  of  them — can  be  the  proper  or  correct  aver- 
age to  draw  for  the  purpose  we  are  putting  it  to.  Having  three 
ways  of  combining  or  averaging  a  thing's  particular  exchange- 
values  into  its  one  general  exchange-value  in  all  other  things, 
we  have  au  enibarras  de  riches.ses,  and  much  of  our  subsequent 
labor  will  be  to  decide  between  these  different  forms. 

§  5.  Now  our  purpose  being  to  compare  the  two  combinations 
or  averages  at  the  two  periods,  we  may  make  this  comparison  by 
dividing  the  formula  for  the  second  jieriod  by  that  for  the  first. 
Here  we  evidently  want  that  form  of  the  expression  for  the 
second  period  which  brings  its  contained  particular  exchange- 
10 


146  MATHEMATICAL  FORMULATION 

values  iuto  relation  with  the  particular  exchauge-valucs  of  the 
first  period.     The  comparison  then  Avill  be  as  follows  : 


^^^^      l{a'{aA)  4-  b\bB)  +  c'(cO)  + } 


Mn.  1 ,    , ,_ 


(^,  0 


-  {{aA)  +  (bB) -^  (cC)  + I 


n 

1 


1 


+  TTTTITn  +  T77T7T~N  + 


Mo,  _  n\a'{aA)  ^  b'{bB)  ^   c'{cC) 

Mr 


(3,   0 


01 


Ir    1  1  1 


^  ~  "T>l)-(65)-(cC)-  *  ^   '^ 

Here  we  have  the  formulse  desired  for  the  variation  of  the 
general  exchange-value  of  M,  but  in  forms  more  complex  than 
necessary.  At  the  first  period  the  exchange-values  {ctA),  {bB), 
(gC), are  each  equal  to  the  exchange- value  of  one  money- 
unit.     The  exchange-value  of  the  money-unit  we  take  for  the 

unit  of  exchange- value.     Therefore  (^aA)  =  (bB)=  (cC)= 

=  1 .  Substituting  this  value  of  the  terms  in  the  denominators 
in  each  of  the  above  equations,  tliey  all  reduce  to  1,  thus  : 

^01=^^^(1  +  1  +  1  +  )=y^nl)=l, 

1 1 1  _ 

'"~  1/111  \~  l/n\  "1-^' 

n(l  +  l  +  l  + j       n[l) 

Mo,  =  ^/r- 1  •  r  rr:::.  =  yv  =  i. 

Therefore,  all    these  denominators    may    be  dropped.      In   the 

numerators  the  terms,  if  written   (ci'ajl),  (b'bB),  (c'cC),  , 

are  also  each  e(|ual  to  the  exchange-value  of  the  money-unit  at 
the  secoiul  period,  and  so  tiic  nunierators  also  reduce  to  1 — only 
this  is  a  difierent  unit.  But,  of  course,  we  want  the  same  unit 
to  be  employed   in  all  the  terms  throughout  the  equations,  and 


OF  EX('iiA.\(ii:-VAT.rK  tm:latioxs  147 

the  unit  wanted  is  the  unit  (■ni|)l()y(Ml  in  tlic  dcnouiinators — the 
excluvngr-valuc  of  tlic  nioucy-unit  at  the  first  or  basic  period. 

Now  in  the  numerators  the  terms  (ciA),  (bB),  (cC), ,  though 

no  longer  equal  to  the  money-unit  at  tlie  second  period,  still  are 
expressit)ns  for  the  particular  exchanoe- values  of  M  at  the  first 
period,  and  so  are  still  equal  to  the  old  unit.  Therefore,  as  we 
em})loy  this  luiit  throughout,  these  terms,  each  =  1,  may  be 
dropped  from  the  numerators  also,  and  the  equations,  thus  freed 
from  useless  symbols,  reduce  to  the  folhnving- : 

^  =  i(^''  +  ^'  +  ^'  + ),  (4,0 


M,,  1 


Mo,      1/1 


(-1,0 


M-l/^'-''-^'   ■■■■•■'  (-^'^) 

Here  we  have  the  Avorkable  forms  of  the  formulse  desired. 
In  these  forms  the  formulae  each  express  one  of  the  mathematical 
averages  of  the  variations,  from  the  first  to  the  second  period,  of 

the  exchange-values  of  M  in  [A],  in  [B],  in  [C], ,  that  is, 

of  the  variations  of  all  the  particular  exchange-values  of  the 
money-unit  in  every  other  class  of  things.  This  appears  to  be 
as  it  should  be  ;  for  a  thing's  variation  in  general  exchange-value 
in  all  other  things  would  seem  to  be  made  up  of  the  variations 
of  all  its  particular  exchange-values,  and  therefore  to  be  an 
average  of  some  sort  between  all  the  particular  variations.  But 
we  reached  this  position  only  through  really  comparing  two  dis- 
tinct averages,  in  each  of  which  the  hy])othesis  was  that  the 
classes  are  equally  important — i.  e.,  equally  important  at  each  of 
the  periods.  And  in  both  these  averages  we  used,  in  obedience 
to  the  princi})le  stated  in  Chapter  IV.,  Sect.  V.,  §  9,  the  same 
number  of  classes — the  same  n.  If  we  had  used  a  different 
number  of  classes  in  each  averaging,  we  could  not  have  made 
this  reduction  to  a  common  averaging.  Whether  we  could  make 
this  reduction  if  the  classes  were  unequally  important  at  each 
jjeriod  singly,  but  were  equally  important  over  both  the  periods 


148  MATIIK.MATICAL    FORMULATIOX 

together,  is  questionable.  Perhaps  it  may  he  proper,  perliaps 
not.  By  most  Avritcrs  on  the  subjeet,  however,  the  (piestion  has 
hardlv  been  raised.  The  fine  distinctions  as  to  tlie  periods  when 
the  weiu'htino;  is  to  be  measnred  have  not  been  noticed.  Gen- 
erally  the  assumption  has  been  made  that  we  can  average  the 
variations  of  the  particular  exchange- values  using  a  single  weight- 
ing, even  though  the  weights  of  the  two  periods  separately  are 
different.  We  may  for  the  present  ignore  this  question,  and  as- 
sume that  we  are  justified  in  using — or  are  dealing  Avith  cases 
which  [)ermit  us  to  use — even  weighting  in  an  averaging  of  the 
variations  of  the  particular  exchange-values. 

§  6.  The  averages  of  the  variations  may  be  represented  in 
still  another  way.  Instead  of  representing  the  numbers  of  mass- 
units  of  the  A'arious  classes  purchasable  by  31  at  the  second 

period  by  aUi,  h'b,  c'c,  ,  let  us  represent  them  by  a.^,  6.,,  c^, 

,  in  distinction  from  which  we  shall  represent  the  numbers 

purchasable  at  the  first  period  by  a^,  h^,  Cj, In  this  nota- 
tion «,„  6.,j  c.„  ,  taking  the  ])laee  of  <i'a,  h'b,  e'c,  ,  are 

a., 
equal  to  them  respectively  ;  wherefore,  as  a.^  =  «'<q,  —  =  "' , 

and  similarly    '  =  1/ ,    '  =  r' ,  and  so  on.      In  other  words,  the 

1    .      ,  '.,«.,     b.,     c, 

variations  are  represented,  m  tlie  new  notation,  l)v  — ,    .",      , 

«j    Oj    Cj 

Therefore  these  new  terms  may  take  the  places  of  a' ,  b' ,  c' ,  

in    the  formula  4,  1-3,  and  we    have  equally  good    and    valid 

formulae  in  the  following  : 


Ma-y       1  {a.,      b.,      (',, 

— ^  =   -    -^  -f  -^  -f  ^  + 

Mo,  1 


n' 


lOh  +  ih  +  'h 


\  (5,   0 


Mn.        ^      a,    b,    c.  •  ^   '     ^ 


§  7.   Any  of  these  averages  b(>ing  adopted,  if  the  result  in  any 
comparison  of  two  periods  is  larger  tlian  unity  in  a  certain  pro- 


OK  i:x('HAX(;K-\Ai,ri;   i;i;i,a'ii(>ns  149 

portion,  this  means  tluit  tlic  exc'han(>:e-valuo  of  money  in  all 
other  things  is  indieated  1)V  this  kind  of  averagino-  to  have  risen 
in  that  proportion  (/'.  //.,  if  the  result  is  1 . 1  '>,  tlie  rise  is  by  1  •")  per 
cent.) ;  if  the  result  is  less  than  unity  in  a  eertain  jn'ojiortion, 
the  exchange-value  of  money  is  indicated  to  have  fallen  in  the 
same  proportion  (e.  g.,  if  the  result  is  .85,  the  fall  is  by  15  per 
cent.) ;  and  if  the  result  is  unity,  the  exchange-value  is  indicated 
not  to  have  varied,  /.  e.,  to  have  remained  constant. 

If  the  variations  of  all  the  particular  exchange-values  of  M  are 

the  same,  so  that  c//=  6':=  c'=  =  r,  m-  -J ?_  -' — ,. 

so  that  this  one  quantity  can  stand  for  them  all,  the  results  of 
all  the  averages  are  the  same,  and  are  r,  as  shown  by  the  fol- 
lowing : 

Mo,     1.     .       ,       ,  ,1 


J/oi      n. 


=  -  (r  +  y  +  /•+  )=     (///•)  =  /•, 


// 


Mo. 1 __]^ 1^_ 

-^^»-vi+i+i+ )  'cyy" 

n  \r       r      r  J      )i  \r  J       r 

Mo, 


31   -]/''"'  '^ -\/    '■  -'■ 


Thus  all  these  results  agree  with  the  knowledge  we  already  pos- 
sess that  if  a  thing  varies  in  all  its  particular  exchange-values 
in  a  certain  proportion,  it  varies  in  exchange-value  in  all  other 
things  in  the  same  proportion  (see  Proposition  XVII.).  Un- 
fortunately, therefore,  that  piece  of  knowledge  does  not  help  us 
to  decide  between  these  methods  of  averaging.  Of  course,  if 
there  is  no  variation  in  any  of  the  particular  exchange-values, 

so  that  a'  =  h'  =c'  =  =  1,  or  ^2_^  =  f?_ _  i^    the 

result  in  all  cases  is  1,  meaning  that  there  is  no  variation  in 
the  general  exchange-value  (in  accordance  with  Proposition 
XXVII.). 

But  if  there  is  variation  in  at  least  one  particular  exchange- 
value,  or  a  variatk)n  in  at  least  one  particular  exchange-value 
divergent  from  all  the  other  variations, — if  there  is  the  slightest 


150  MATIIEMA  riCAI,    FOKMri.ATIOX 

irreffularitv, — tlie  ivsiilts  in  the  three  kinds  of  nverau'ino-  alwavs 
differ  from  each  other,  lietween  the  three  mathematical  aver- 
ag:es  tlie  matliematical  rehitions  are  well  known.  If  tlie  varia- 
tions hai)]>en  to  be  sneh  that  the  geometi'ie  averag'e  indicates 
constancy,  the  arithmetic  averai;:e  will  always  indicate  a  rise,  the 
harmonic  a  fall.  If  the  g-eometric  average  indicates  a  rise,  the 
arithmetic  will  indicate  a  greater  rise,  the  harmonic  a  lesser  rise 
(perhaps  constancy,  perhaps  a  fall).  If  the  geometric  average 
indicates  a  fall,  the  arithmetic  will  indicate  a  lesser  fall  (perhaps 
constancy,  perhaps  a  rise),  the  harmonic  a  greater  fall.  The 
geometric  average  is  always  between  the  arithmetic  above  and 
the  harmonic  l)elow.  In  fact,  when  there  are  only  two  eqnally 
important  terms  used,  the  geometric  mean  is  the  geometric  mean 
between  the  arithmetic  and  the  harmonic  means  ;  and  in  other 
cases  the  geometric  average  is  near  that  mean." 

It  is  ])lain  that  in  none  of  the  methods  can  constancy  be  in- 
dicated if  there  is  only  a  variation  of  one  j)articular  exchange- 
value  of  money,  or  if  all  the  variations  ar(>  in  the  same  direc- 
tion ;  but  only  in  the  case  of  opposite  and  com])ensatory 
variations.  This  also  agrees  with  what  we  already  know  (see 
Propositions  XX.  and  XXVIII. ).  Given  a  single  variation, 
however,  each  of  the  methods,  in  order  to  indicate  constancy,  re- 
quires a  different  compensatory  variation — the  arithmetic  the 
greatest  if  the  coni])ensatory  variation  l)e  a  fall  and  the  smallest 
if  a  rise,  the  harmonic  tlu;  smallest  if  a  fall  and  the  greatest  if  a 
rise,  the  geometric  ahNays  an  intermediate. 

II. 

§  1 .  It  is  easy  merelv  to  inti-ochiee  nne\'en  weigliting  into  tlie 
formulae.  We  only  have  to  remember  what  uneven  weighting 
means.  Uneven  Aveighting  means  that  one  class  is  larger  than 
another,  that  is,  contains  more  individuals  ;  while  classes  evenly 
weighted  are  supposed  to  contain  an  equal  number  of  individ- 
uals. What  these  individuals  are  we  need  not  liere  again  dis- 
cuss. Now  if  a  class  contains  twice  as  many  such  individuals 
as  another,  it  virtually  consists  of  twice  as  many  classes  as  the 

11  See  Appendix  A  VI.  g§  1-5. 


OF    EXCHANGE- VALUE    JJKLATIONS  151 

other;  and  if  we  treat  the  other  as  one  class  we  may  treat  this 
as  two  chisscs.  The  donblc  nnmber  of  individnals  which  are 
conihined  into  tlie  lirst  cU\ss  are  so  combined  into  one  instead  of" 
into  two  chisses  only  because  they  all  have  the  same  name  and 
the  same  excliauge-valne.  They  are  one  class  nominally  ;  but 
logically  they  are  two  liom(»nymous  classes — compared  with  the 
other  as  one  class.  Thus  suppose  the  class  [B]  is  twice,  and  the 
class  [C]  thrice,  as  large  as  the  class  [A];  the  state  ofthings  then 
is  the  same  as  if  we  had  two  classes  both  called  [B] ,  in  both  of 
Avliieh  M  has  always  the  same  exchange-value,  and  three  classes 
all  called  [C] ,  in  all  of  which  M  has  always  the  same  exchange- 
value.  If  this  be  supposed  to  be  the  state  of  things  at  the  first 
period,  then  at  the  first  period  the  formulae  would  be  as  follows  : 

3Ioi=  ^,  \  {(lA)  +  {bB)  4-  {bB)  +{cC)  +  {cC)  +  {cC)  + to  n^'' terras} 

=  i  I  {aA)  +  2{bB)  +  3{cC)  + to  n  termsf,  (6,  i) 

1 

"^^^''=1(1,1         1,1,1,1.  ^      „^  ) 

-7.  -^  — r  +  Cv  +  Td  "T  -rr  +  -p,  +  -79  + to  71^'^  terms  \ 

n"  [  aA       bB     bB      cC       cC       cC  J 

1 

^~~2         3^  f'  (6,2) 

«2  +  6£  +  c^+ tonterms    | 

31 01  =  "i^ {aA)  ■  {bB)  ■  {bB)  ■  {cC)  ■  {cC)  ■  (cC) to  n''  termF 

=  '"'i/{aA)-  {bB)-^-  {eC)^ tow  terms;  (6,  s) 

in  which  n"  represents  the  number  of  the  virtual  classes,  or  n 
enlarged  to  cover  all  the  coefficients  or  exponents. 

If,  then,  at  the  second  period  the  same  relative  conditions  per- 
sisted, in  spite  of  variations  in  the  particular  exchange-values, 
we  should  have  formuhTe  similar  to  these,  with  addition  only  of 
the  symbols  indicative  of  the  variations.  Then  the  comparison 
is  easy.  We  may,  however,  represent  the  weighting  more 
generally  by  employing  algebraic  symbols  throughout.  Let  a 
represent  the  weight  of  [A] ,  b  the  weight  of  [B] ,  C  the  weight 
of    [C],  and  so  on,  all   these  weights  being  supposed  to  be  the 

same  at  both  periods  ;  and    let  n"  =  a  +  b  +  C+-f-  to  n 

terms,  the  same  classes  of  course  being  used  in  each  of  the 
averages  conijiared.      Then  we  have 


152  MATIiE.ArATIOAI.    FORMULATIOX 

,r  „{B.a'{aA)  +  'b  b'{bB)  +  C  c'{cC)-{- ton  terms  } 

M    ~  1                                                                          '^  '  ^ 

^'  "„{a(«^)+b(6^)  +  c(cC)  + to  n  terms} 


1  f     a  b  c  ) 

T/i^77 — 1T+///;  />\+   //  /-x  + to  n  terms  ^ 

i"{a'{aA)      b'{bB)     c\r(J) j_  ^    ^^^ .. 


Mo^ 


(^  +  (/.7>0+(cC')  + t<>.  terms 

J/o2_i/^{fi'M)}^-  {b'{bB)}^-  {c'(cC)}' tonterms 

"^^^1  l/ {aAY~IbB)*'  ■  {eCf to  n  terms 

In  all  these,  agaiu,  the  denominators  reduce  to  unity  ;  and 

also  the  expressions  (aA),  (bB),  (cC),  in   the  numerators 

are  the  same  units.     Therefore  we  have 

31         1 

-^^'-  =    ,,  (a  rt'  +  b  6'  +  C  e'  +  to  n  terms),      (8,  i) 

mI  ^ 1         

Mox       1  /a      b      c  \ 

"^  I L    —   J _L   fn    11     forme  \ 


n"  \a 


/a       b       c  \' 

V?  +  /7  +  ;^  + ^'' "  ^'^^v       ^^'  '^ 


Mm  _  j^/^^,a  .  ^/b  .  ^,c  .  ^  nterms.  (8,  3) 

Or  if  we  prefer  the  other  system  of  notation  to  represent  the 
variations,  we  have 

3£02  1      /„   «2      ,      1-^2      ,      „  «2 


=  .H'^;  +  ^^''''f+  tontermsj,    (9,.) 

,,  (  a  -^  +  b  7-'  +  C  -  +  to  n  terms  ) 

/'V    o,  6,          (-2  y 

Here  again  are  formulae  representing  averages  of  tlie  variations 
of  the  particulnr  excliaiige-values.     The  reductions  by  which  they 


Mor 

31,  _ 

3Ifj,        1    /    «.       .  b 


M,, 
Mo, 


OF    KXCIIAXGE-VALrE    KELATIONS  153 

are  obtained  suppot^e  that  tlie  weights  are  the  same  at  each  pci'iod 
separately.  But  sicnerally  in  practice  the  weiglits  are  different 
at  each  period  se})arately.  Still  if  we  are  to  average  the  varia- 
tions, we  must  emph)y  only  a  single  system  of  weighting — for 
we  cannot,  for  instance,  have  a  variation  of  two  classes  of  [A] 
and  a  variation  of  three  classes  of  [.\  ] .  We  must  have  either 
a  variation  of  two  classes  of  [A] ,  or  a  variation  of  three  classes 
of  [A],  or  a  variation  of  a  number  of  classes  of  [A]  that  is  a 
mean  between  the  numbers  of  classes  of  [A]  at  each  of  the 
periods  separately,  as  examined  in  the  preceding  Chapter.  For 
the  present  we  may  assume  that  this  course  is  a  proper  one,  and 
that  one  of  these  weightings  is  the  proper  one.  We  may,  then, 
continue  to  represent  the  weight  of  [A]  by  a,  and  that  of  [B] 
by  b,  and  so  on,  and  their  sum  by  n",  in  Avhatever  way  these 
weights  be  calculated. 

The  averages  agree  with  much  that  we  have  already  discov- 
ered concerning  variations  of  exchange- value,  and  disagree  with 
nothing  we  yet  know.  Thus  it  is  plain  that  in  any  of  these 
formuhe  a  change  of  weighting  without  any  change  of  exchange- 
values  causes  no  change  in  the  result ;  for  then  the  terms  a' ,  b', 

c'   ,  or  —  ,  7-^,   — , ,  are  all  1,  and  its  does  not  matter  how 

large  or  small  a,  b,  C, be,  the  results  are  always  1,  indicating 

constancy  in  the  general  exchange-value  of  money.  This  is  in 
agreement  with  Propositions  XXVII.  and  XLIV.  Also  if  tlie 
variations    are    all    alike,   so    that    a'=h'  =  c'  =  =  r,   or 

«..       ^>.,      '".>  Ill  -1         1 

—  =  ~=  —= =  r,  no  matter  what   be  the  weights,  the  re- 

«i       ^      ^'i  '  _  .  .  / 

suits  are  always  r,^  in  agreement  with  Propositions  XVII.  and 

XLV. 

In  all  these  formulae  the  results  are  the  same  whether  we  use 

the  weighting  a,  b,  C,  ,  with  n"  =  a  -|-  b  -f-  C  -f-  to  n 

terms,  or  whether  we  use  /•  a,  /■  b,  /•  C, ,  with  n"  =  /•  a  +  /•  b 

-|- /■  C to   /(   terms. ^     This  means  that  it  is  indifferent  how 

large  or  small   be  the  weights,  provided  they  always  maintain 

^  Cf.  Appendix  A  I.  ^^'),  <;,  H.  iJ  (;,  III.  iJO,  V.  iJ,'). 
~Cf.  Appendix  A  I.  ji  11. 


l."')4  MATHEMATICAL    FORMULATION 

the  same  relative  sizes.  Therefore  the  ^veiu'llt^  may  he  any 
numbers,  larger  (»r  smaller  than  unity,  integral  or  fractional. 
It  is,  then,  not  necessary  that  any  of  tlie  classes  should  be  taken 
as  a  unit  class  like  [A]  in  the  above  numerical  example,  al- 
though our  subsequent  calculations  might  be  slightly  simpler  if 
this  were  done  and  the  other  \veights  reduced  accordingly.  In 
this  case  the  coefficient  of  the  unit  class,  and  of  all  the  classes 
equal  to  it,  namely  1,  must  be  counted  in  the  composition  of  the 
term  /;/'. 

§  2.  It  may,  liowever,  turn  out  that  this  course  of  averaging 
the  variations  on  a  single  weighting  is  not  right,  no  weighting 
being  discoverable  that  Avill  in  all  cases  yield  a  demonstrably  cor- 
rect result.  If  this  l)e  so,  we  shall  not  be  able  to  make  the  re- 
duction of  the  comparis(Mi  of  the  two  averages,  one  at  each  period, 
to  a  single  average  of  the  variations,  but  shall  have  to  retain  the 
comparison  of  the  tw'o  averages.  This  constitutes  what  Me  have 
called  the  method  of  double  weighting.  In  our  formuhe  we  shall 
have  to  distingush  a  into  a^  and  a.„  b  into  b^  and  b.„  and  so  on, 
representing  the  weights  at  the  first  and  second  periods  respec- 
tively ;  and  consequently  also  n"  into  n^'  aud  »./',  although  n 
must  remain  undivided.  Then  the  formuhe  will  become  the  fol- 
lowing : 

,,  — -,,\2i,2(i\aA)  +  hib^ibB)  ■\-  c.iC^{cC)  + to  n  terms  \ 

-j~=^- ,     (10,    i) 

^^  ,,Jai(«-l)+b,(ii3)-f  C,(cC)4- tontermsl 

1 

1    [     a.>      .      b.,  C2  I 

iJTw 1  ~  '         ^    '  '' 

^^/  ^  7~ir\  +  VXrk  +  T^  4- to  n  terms  [ 

11^"  \\a.A)       [bB)        (cC)  J 

Mm      'y^U'{a^)\^'-  \b'{bB)\^'-  \c'{cC)]^' to  n  terms 

1/    [aA)^'  ■  (hB)*''  ■  {cCf' to  n  terms 

But  here  the  denominators  again  all  reduce  to  unity,  aud  may 
be  omitted  ;  and  if  in  the  numerators  the  terms  (aA),  {bB), 
(cC),  be  conceived  as  referring  to  the  exchange-values  of 


OF    KXCIIANfiK-VALUH    RELATIONS  155 

the  money-unit  at  the  first  ])cri()(l,  and  tlierefore,  as  mere  units, 
be  omitted,  the  fornmhe  ix'chice  to  ex[)ressions  for  the  averages 
of  the  variations  Avitli  the  weigliting  of  the  seeond  period — whieh 
is  absunh    And  il",  then,  we  conceive  of  the  terms  n'((iA),  h'(bB), 

g'(cC), as  referring'  to  the  exehange-vahies  in  tlie  money  of 

the  second  period,  whereupon  they  each  become  e<pial  to  unity, 
the  numerators  also  reduce  to  unity,  and  all  these  formulae  are 
useless. 

What  we  need  is  tliis.      The  terms  a,  b,  c,  ,  and  a'a,  b'b, 

c'e, ,  or  better,  ((^,  b^,  c,, and  a.„  b^,  r.^, ,  express  the 

numbers  of  times  the  money-unit  is  more  valuable  than  certain 

mass-units  of  [A]  ,  [B] ,  [C]  , at  the  first  and  at  the  second 

period  respectively.  Therefore  we  want  to  average  these  ratios. 
But  to  average  them  ])roperly  Ave  need  to  know  how  often  they 
are  repeated  at  each  period.  Let  .i\  and  x^  represent  the  num- 
bers of  times  the  mass-unit  of  [A]  is  repeated  at  each  period, 
that  is,  the  numbers  of  these  mass-units  that  have  been  })roduced 
or  consumed  within  these  periods  ;  and  let  y^  and  ;/.„  z^  and  z^, 

represent  the  numbers  of  the  mass-units  of  [B],  [C], • 

And  let  n/  =  ,i\  -\-  i/^  -j-  z^  -\-  to  n  terms,  and  nj  =  .r,  -|-  y^ 

4-  ^2  + to  n  terms.      Then  the  desired  formulae  are 

1 

If         ,77(-V'2  +  3/2^2  +  -^2  + to  ^  terms) 


/(■*'i«i  +  .Vi^i  -t-  .-h''i  +  to  n  terms) 


_  »/(.?-/^o  +  y.J),  -\-  zj^,  -f-  to  n  terms)        -^^ 

""/(•^*i«i  +  2/1^1  +  -h^i  + to  n  terms)  ' 


2 

1 


1   f.v,        I/.,        z,  \ 

(  -^  -j_  f^  -f    -'  -f  to  n   terms  ) 

\«2       ^2       ^'2 / 

1 

—,(—  +  'i^  +  '-  +  to  n  terms  ) 

7//V^'i         ^i  ''1  / 

n,(  —  ^t'-^-  '^  + to  n  terms  I 

'Y'l       '^1 Ci I 

ni-  +  T^  +  '"'  +  to  n  terms  ) 

\n.,      b.,      (\^  ) 


-^02  _  ^ 

Wo,-  1 


156  MATHEMATICAL    BX)RMULATlON 


31  o      ]/    ^^2^' '  ^2' '  ^2"' ^  '"'  terms 


31 


^^      I  /   ^i"^'  •  6j'^'  •  Cj"' to  71  terms 


(11,.) 


These  may  also  be  written  more  fully  by  expanding  the  »/s. 
The  first  so  treated  becomes 

if 02  ^  ayt^  +  i/^b^  +  -Va  +  ,  -^^  +  3/1  +^1+ 

il/01      .Tjff^  +  7//;^  -f  2jCj  +  ■  .r^  +  3/2  4-  .^2  + ' 


which  is  the  general  fornuila  for  the  method  of  double  weighting 
with  the  arithmetic  average. 

But  here  au  important  remark  nnist  be  made.  It  is  evident  that 
in  all  these  last  formulae  the  results  will  be  variously  aifected  by 
the  sizes  of  the  mass-units  we  happen  to  choose.  For  if  we  choose 
large  mass-units,  the  numbers  of  times  they  occur  at  each  period 

(the  expressions  .r,  //,  z, )  and  the  numbers  of  times  the  money- 

imit  is  more  valuable  than  they  (the  expressions  a,  b,  c, )  are 

smaller  than  the  numbers  of  times  smaller  mass-units  would 
occur  and  than  the  numbers  of  times  the  money-unit  would  be 
more  valuable  than  smaller  mass-units.  In  some  of  the  terms, 
as  in  the  parts  Avithin  brackets  in  the  second  formula,  these  two 
changes  neutralize  each  other ;  but  not  in  others,  and  never  in 
the  terms  n^'  and  n./,  except  only  if  all  the  mass-units  be  altered 
in  the  same  proportion.  But  if  the  mass-units  be  variously  and 
indiscriminately  chosen,  the  terms  n^'  and  n/,  not  to  mention 
others  in  some  of  the  formulfe,  will  variously  relate  to  each  other, 
and  the  results  will  be  various,  in  a  haphazard  manner.  Hence, 
it  is  necessary  that  some  method  of  selecting  the  mass-units  be 
established,  as  without  such  a  method  also  these  formula;  are 
worthless.  Such  a  method  need  not  be  stilted  in  the  formula, 
and  tlien  the  above  formuke  become  applicable  only  <»n  the  sup- 
position that  the  method  of  selecting  the  mass-units  has  already 
been  found  and  applied.  Or  again  it  may  sometimes  be  possible 
to  introduce  into  the  fornuila  itself  the  method  of  selecting  the 
mass-units.  This  has  actually  been  done  in  some  methods  deal- 
ing with  the  inverted  cases  of  price  measurements.  '^Pliese  we 
shall  examine  later. 


POSSIBLE    ERRORS  157 


III. 


§  1.  Certain  waniiiijis  arc  needed  against  errors  into  whicli 
the  nse  of  mathematical  formnlse  in  tliis  subject  may  lead  us, 
unless  we  be  careful. 

Two  general  principles  are  obtained  from  the  preceding  in- 
vestigations. The  one  is  that  in  cases  when  the  weighting  is 
the  same  at  both  the  })eriods  compared,  it  is  immaterial  whether 
we  average  all  th(>  ])articular  exchange-values  at  the  two  periods 
separately  and  then  find  the  variation  of  these  averages,  or 
whether  we  average  all  the  variations  of  the  particular  exchange- 
values.  The  other  is  that  when  the  weightmg  is  different,  and 
different  weighting  is  used  in  the  averaging,  at  each  period,  we 
can  only  find  the  variation  of  these  averages,  there  being  no  one 
averaging  of  the  particular  variations  that  can  use  double 
weighting.  AYe  have  seen,  however,  that  we  can  draw  an  aver- 
age between  the  t'svo  weightings  ;  and  now  we  might  use  this 
single  weighting  in  averaging  the  particular  variations.  But 
we  shall  find  that  this  last  operation  will  yield  the  same  result 
as  the  comparison  of  the  two  separate  averages  only  in  peculiar 
circumstances.  We  therefore  have  before  us  two  distinct 
methods.  The  one  is  to  compare  the  averages  separatel}'  drawn 
at  each  period.  The  other  is  to  average  the  particular  varia- 
tions. These  two  coincide  only  when  the  weighting  actually  is 
the  same  at  both  periods,  or  under  other  peculiar  conditions. 
Even  in  the  first  case,  when  the  weighting  (reckoned  according 
to  the  relative  total  exchange-values  of  the  classes)  is  the  same 
at  both  periods,  there  might  be  made  averages  at  each  period  on 
other  systems  of  weighting  (e.  g.,  according  to  mass-quantities), 
which  might  be  different  at  each  period,  or  if  the  same  at  both 
periods,  might  be  so  only  in  case  the  Aveighting,  proj)erly  reck- 
oned, is  different  at  each  period. 

Thus  the  subject  is  very  complex.  Running  through  it  all  are 
the  similarity  and  difference  between  comparisons  of  averages  that 
have  varied  and  averages  of  the  particular  variations — a  purely 
mathematical  question,  which  has  never  been  fully  studied  even 
bv  mathematicians,  let  alone  economists.      Hence  it  has  been  nee- 


158  MATHEMATICAL    FORMULATION 

essary  in  this  work  to  make  such  a  matliematical  study,  to  scryc 
as  a  foundation  for  the  yarious  applications  of  the  mathematical 
princii:>les  which  we  shall  have  to  make  in  dealino;  with  the 
averages  of  exchange-values  and  of  prices,  and  with  their  varia- 
tions.    This  study  is  added  as  Appendix  A. 

There  are  many  difficulties  engendered  by  the  many  complexi- 
ties in  our  subject,  ^vhich  Ave  shall  meet  from  time  to  time.  But 
there  are  some  simple  errors  into  which  we  might  easily  slip  at 
the  commencement,  and  into  which  many  persons  have  slipped. 
Some  of  them  are  even  suggested  by  the  second  method  of  no- 
tation above  given — a  method  Avhich,  in  its  general  feature  of 
distinguishing  the  exchange-values  (or  again  the  prices)  of  the 
same  classes  at  the  different  periods  only  by  different  subscript 
numbers  ajipeuded  to  the  same  letters,  is  the  simplest  and  the 
most  commonly  used.  Against  the  errors  so  suggested  it  is  well 
here  to  give  w^arning.  These  occur  even  when  the  weighting 
is  the  same  at  both  periods — or  when  we  are  attempting  to  use  a 
single  weighting  in  averaging  variations.  Confinement  may  be 
made  for  the  present  to  such  cases. 

§  2.  A  general  error  is  this.  Because  in  these  cases  the 
method  of  averaging  the  variations  properly  agrees  with  the 
method  of  comparing  the  averages,  and  because  the  former  is  the 
simpler  method,  we  might  be  tempted  to  say  of  exchange-values 
what  Jevons  said  of  prices,'  that  there  is  no  such  thing  as  an 
average  of  them,  but  only  an  average  of  their  variations.  In- 
deed, as  all  the  particular  exchange-values  of  anything  at  any 
time  are  always  equal,  not  only  the  average  of  the  ])aili('ular 
exchange-values  of  money  properly  weighted  at  the  first  period 
is  always  unity,  or  lilioi?  ^>"t  at  the  second  period  also  the  aver- 
age of  the  ])articular  exchange-values  of  money  properly  weighted 
will  also  be  unity  in  the  new  unit,  31,,.^.  Still  there  may  be  a 
different  average  of  the  exchange-values  at  the  second  period  in 
the  unit  of  the  first  period  ;  which  is  just  what  we  want,  both 
because  we  ought  to  conduct  the  whole  operation  on  the  same 
unit,  and  because  it  is  convenient  that  the  average  at  the  first 
period,  with  Avhich  that  at  the  second  is  com]iarcd,  should  always 

1 B.  22,  p.  23. 


POSSIBLE    ERRORS  159 

be  1,  as  it  forms  the  denominator  in  the  expression  for  the  com- 
parison. The  proeess  then  is  the  following  :  At  the  first  period 
we  construct,  as  it  were,  a  level  of  the  particular  exchange-values 
of  money.  This  level  breaks  u])  at  the  second  period  into  an 
undulating  line,  of  the  various  heights  of  which  an  average  may 
be  drawn  and  compared  with  the  uniform  height  of  flie  first 
level.  This  ojicration  evidently  gives  both  an  averaging  of  the 
variations  and  a  variation  of  the  average  at  the  second  ])eriod 
from  the  level  or  average  at  the  first.  It  is  difficult,  in  fact,  to 
conceive  of  an  average  of  variations  without  admitting  an  aver- 
age of  the  starting  and  of  the  finishing  points.  But  the  averages 
do  not  exist  in  nature,  nor  are  the  positions  given  upon  which 
averages  merely  need  to  be  drawn.  The  Avliole  operation  has  to 
be  constructed,  and  error  will  ensue  unless  it  be  constructed  in 
the  right  way.  Our  guiding  consideration  must  be  that  the 
variation  of  the  averages  must,  in  these  cases,  give  the  same  re- 
sult as  the  proper  average  (of  the  same  mathematical  species)  of 
the  variations.  This  agreement  is  always  obtained  if  we  con- 
struct the  average  at  the  first  period  upon  equal  terms,  so  that 
the  average  itself  is  equal  to  them,  and  then  use  the  same  weight- 
ing in  both  methods. 

But  the  agreement  is  also  ol)tainable  by  comparing  separate 
averages  at  each  jieriod  without  the  average  at  the  first  period 
being  so  constructed  upon  equal  terms.  This  occurs  always  in 
the  geometric  average  if  we  use  the  same  weighting  in  both 
methods  ;  but  in  the  arithmetic  and  harmonic  averages  never  if 
we  use  the  same  weighting,^  and  only  if  we  use  different  weight- 
ing in  each  method,  a  definite  relation  existing  between  those 
which  have  to  be  employed  to  get  the  same  result.  For  in  the 
arithmetic  and  harmonic  averaging  the  inequality  of  the  terms 
themselves  at  one  of  the  periods  forms  part  of  the  weighting  in 
the  comparison  of  the  separately  made  averages,  but  has  no  in- 
fluence on  the  weighting  in  the  averaging  of  the  variations ; 
wherefore,  unless  it  is  allowed  fi)r  in  the  fi)rmer,  an  ap])arent 
similarity  in  the  weighting  will   really  l)e  a  ditference,  and  real 

-  Except  when  all  tlie  variations  are  exactly  alike,  or  when  there  are  no  varia- 
tions. See  Appendix  A  I.  ^  6.  As  these  cases  rarely  happen,  and  as  we  have  no 
control  in  regard  to  them,  they  are  not  of  much  importance. 


160  MATHEMATICAL    FORMULATION 

similaritv  can  hv  obtained  only  by  omployino-  different  a])])arent 
wei^htiug-. 

Now  if  we  had  at  the  start  employed  the  form  of  notation  kiter 

introduced,  namely  that  of  using  a.^,  b.,,  e.,, for  the  numbers 

of  the  mass-units  of  the  classes  j)urchasable  with  the  money-unit 
at  the  Second  period,  we  might  have  fiUen  insensibly  into  the 
treatment  of  comparisons  of  averages  employing  even,  or  what 
we  thought  to  be  the  proper  uneven,  weighting,  and  to  have 
supposed  that  we  were  then  employing  even,  or  the  same  proper 
uneven,  weighting  in  averaging  the  variation  ;  or  if  we  happened 
to  notice  the  divergence  in  the  results,  we  might  have  been  non- 
plussed, and  perhaps  have  given  up  the  subject  in  despair,  as 
some  economists  have  done. 

The  errors  we  could  thus  fall  into  are  different  in  the  differ- 
ent kinds  of  averaging  ;  A\herefore  these  need  to  be  treated  sepa- 
rately. 

§  3.  Arithmetic  averagim/. — Intending  to  employ  even  weight- 
ing at  both  periods  and  in  the  averaging  of  the  variations — or  to 
disregard  weighting,  using  "  unweighted  "  averages — we  might 
represent  the  average  at  the  first  period  thus. 


Mo,  =  ~{{chA)  +  {\B)  +  {c,C)  + 


n 


and  tliat  at  the  second  period  thus. 


Mo,=     Ua,A)  +  {h,B)  +  {e,C)  + 


n 


wherefore  the  comparison  of  these  would  be 

M-liieli  reduces  to 

Mo,  _  {a,A)  +  jb^B)  +  (c,C)  +  • 

M,,-{a,A)-\-{b,B)  +  {e,C)  + ' 

the  variation  of  the  arithmetic  averages  being  the  same  as  the 
variation  of  the  sums.    Both  of  these  comparisons  of  tlie  averages 


POSSIBLE    KRRORS  161 

woitkl  be  perfectly  eorrect,  thoiij^h  meaninijless  witliont  f'ui'tlier 
tivatnient.      Hereupon  wo  mio;lit  have  ar<!:ue(]  that,  as  ji,  />',  (', 

seem  to  represent  unity,  or  merely  to  make  reference  to  the 

classes  [A],   [B],   [C], ,  therefore  tluy  can  he  omitted,  and 

so  we  should  have 

JI„,  _  a.,  +  f,,,  +  e.,+  --^ 

J/,„  ~"  ((^  +  6,  +  e,  + ■ 

This  argumentation,  however,  would  be  wrong,  because  it  is 
not  A,  B,  C, which  are  ecpial  to  unity  (the  exchange- 
value    of    the    m(mey-unit),   but    (aA),    (bB),  (gC), ;    and 

A,  B,  C, do    not    refer    to    the    classes,    but,    taken    thus 

alone,  to  the  general  exchange-values  of  the  mass-units  of  the 
classes.  Then,  again,  if  we  dropped  only  the  denominator  as 
reducing  to  unity,  the  remaining  form 

^;  =  ^{(M)  +  (^3^)  +  (^.o)  + } 

would  be  meaningless,  because  in  the  unit  at  the  second  period 
it  too  reduces  to  unity,  and  it  contains  no  indication  of  the  value 

of  the  terms   {u.^A),  (b.^B),  (c.^C), in  the  original  unit  Mqi. 

We    do,    however,    know    that    a^A  =  h^B  =  c^C=   =  1? 

wherefore  A  =      ?  ^  =    ,  j  C=      ,  and   so  on.      Substituting 

(t^  Oj  Cj 

these,  we  get 

Mqo        1/a,    ^     b,        c. 


U +?:  +  ,.;  + > 


which  is  the  correct  form  already  obtained,  (5,  i). 
Not  only  the  argumentation  by  which  the  formula 

M^o  _  a,  +  b.^  +  c.^+  

J/,,1  ~  a,  -I-  &!  -f-  c,  -I- 

was  obtained,  but  also  this  formula  itself,  is  wrong.  By  saying, 
a  formula  is  wrong  we  mean  it  is  not  universally  right,  although 
it  may  be  right  on  some  occasions.  This  formula  is  wrong  be- 
cause it  does  not  universally  carry  out — and  in  ])ractice  is  rarely 
11 


162  MATHEMATICAL    FORMULATION 

likely  to  carry  out — tlio  idea  Ave  intended  to  carry  out,  namely 
to  employ  even  weighting.  For  (as  is  more  fully  shown  in 
Appendix  A,  II.  §  7)  this  formula  really  contains  a  special 
system  of  weighting  in  the  case  of  averaging  the  variations,  as 
it  agrees  with  the  fornuila 

Mo2       1   /    «.,   ,   ,    fi, 


V''^  +  ^^6:+'''^+ to.^ermsj, 


(in   which  n"=  a^  -\-  b^-\-  c^^  -}-  to   n  terms),  which   is  the 

formula  for  averaging  the  variations  with  the  weights  o^  for  [A], 
/>j  for  [B],  c^  for  [C],  (so  that  there  would  be  even  weight- 
ing here  only  in  case  it  happened  that  «j  =  6j  =  c^  ^  ). 

But  we  have  probably  chosen  these  terms  at  haphazard,  that  is, 
without  paying  regard  to  the  importance  of  the  classes :  we 
have  sought  merely,  in  the  different  classes,  of  the  usually  em- 
ployed mass-units  the  numbers  that  happen  at  the  first  period 
to  be  equivalent  to  the  money-unit.  The  weighting  for  the  vari- 
ations which  we  are  really  using  is,  then,  haphazard  weighting, 
and  the  whole  process  is  absurd. 

Xo  reliable  result  can  be  obtained  in  this  way,  and — which 
proves  it — an  indefinite  number  of  different  results  may  be  so 
obtained.  This  may  be  shoA\n  by  an  example.  .Suppose  the 
money-unit  purchases  two  quarters  of  wheat  at  the  first  period 
and  at  the  second  three,  at  the  first  four  bales  of  cotton  and  at 
the  second  two,  at  the  first  five  tons  of  coal  and  at  the  second 
still  five  ;  and  suppose  Ave  are  trying  to  measure  tlie  exchange- 
value  of  moncA'  in  these  three  other  thino^s  onh'.  We  mio-ht  then 
be  tempted  to  represent  the  aA'crage  exchange-value  of  money  at 
each  period,  and  consequently  the  variation  of  the  a\'erages,  in 
the  following  way, 

if„2_K3  +  -2  +  5)      3  +  2  +  5       111 

377,     W+  ^  +^  ~  ■^+'i  +  •'  °  11  =  *'■■'*-''"'• 

Avhich  indicates  a  fall  of  !).09  percent.  Now  if  it  had  happened 
that  Avheat  had  been  measured  in  bushels  instead  of  in  (piarters, 
the  exchange-value  of  the  money-unit  at  the  first  j)eriod  wouhl 
have  been  ('(pial  to  that  of  sixteen  bushels  of  wheat  and  at  the 


I'<).S,SIJ5LK    ERRORS  163 

second  to  that  of  twenty-four.  Then  if  tliese  titrurcs  liad  been 
used,  we  might  have  rejji'esented  the  aNerages  at  each  period 
and  the  comparison  as  follows, 

M^,^  K--^4  +  2  +  5)  ^  24  +  2  +  .)  ^  31  ^ 
Mo,       i(l<i  +  -^+">)       16  +  4  +  0       25 

which  indicates  a  rise  of  24  per  cent. 

The  reason  for  the  differenee  in  the  results  is  plain.  It  is 
that  in  the  latter  case  a  great  deal  more  importance  or  weight  is 
attached  to  the  rise  of  the  wheat  than  to  the  fall  of  the  cotton 
and  to  the  constancy  of  the  coal,  whereas  in  the  former  case 
there  was  more  importance  attached  both  to  the  fall  of  the  cot- 
ton and  to  the  constancy  of  the  coal.  Now  if  even  weighting 
were  employed,  or  intended,  the  proper  measurement  w^ould  be 
in  this  form. 


Mo, 


1/3        2        5\        1/24       2        o\       ^ 
=  3U  +  4  +  5J=3(l(i+4  +  5J=^' 


which  indicates  constancy  (on  the  supposition  that  the  arithmetic 
average  is  the  right  one  to  employ,  the  rise  of  wheat  by  fifty 
per  cent,  being  counterbalanced  by  the  fall  of  cotton  by  fifty 
per  cent.).  *  The  same  result  would  be  obtained  if  avc  put  the 
comparison  m  this  form 

J/,, -1(1  [2]-f  1[4]+1[5])' 

and  used  only  the  outer  eoefficients.     This  is  what  we  did  when 

we  employed  the  algebraic   symbols  a' a,  b'b,  c'c,  for  the 

second  period   and  a,  h,  c, for  the  first,  and  then   counted 

only  a' ,  b' ,  c' ,  .      This  method  gives  only  one  result^ — the 

only  result  which  can  be  given  with  the  employment  of  even 
weighting  for  the  variations.  The  other  gives  as  many  results 
as  there  are  mass-units  that  might  be  employed.  That  these 
various  results  are  given  by  employing  variously  weighted 
averages  of  the  variations,  in  which  the  weighting  is  according 
to  the  number  of  mass-units  purchasable  at  the  first  period, 
may  be  shown  in  the  above  instances  as  follows, 


164  MATHEMATICAL    FORMULATION 

Mo^  -  2  +  4  +  5  ~  11  V^  ^   ._,  +  ^  ^  4  +  ''  ^  ry)  -  ir 


and 

M"o2_24jf^  +  5_  1   /,,.       24 


/,         24        ,       2       _       r>\       31 


Thus,  in  using  the  arithmetic  average,  when  we  want  to  em- 
ploy even  weighting,  Ave  must  divide  the  exchange-vahies  of 
the  money-unit  in  each  class  at  the  second  period  by  its  ex- 
change-value in  the  same  class  at  the  first  period,  then  add  the 
quotients,  and  divide  by  the  number  of  the  classes.  This  method 
may,  for  short,  be  called  the  method  of  dividing  before  adding. 
The  other  would  be  the  method  of  adding  before  dividing.  The 
latter  always  gives  the  same  result  as  the  former,  and  correctly 
carries  out  the  intention,  if  we  first  reduce  the  numbers  of  the 
mass-units  purchasable  at  the  first  period  to  the  same  figure,  pre- 
ferably to  unity,  or  to  100  (by  creating,  so  to  speak,  a  special 
mass-unit  for  each  class).  It  is  Avrong  simjily  to  add  u])  the  piu- 
chasable  numbers  of  any  mass-units  that  happen  to  be  employed. 
In  doing  so,  we  are  really  using  uneven  weighting  for  the  classes 
according  to  the  numbers  of  these  units  which  happen  to  be  pur- 
chasable at  the  first  period.  The  larger  the  mass-unit  which 
happens  to  be  employed,  the  smaller  is  the  number  of  these  units 
purchasable,  and  the  smaller  the  weighting  ;  and  the  smaller  the 
unit,  the  larger  the  weighting.  Yet  in  spite  of  itself,  it  is  con- 
ceivable that  this  method  might  happen  to  be  right.  For, 
though  intending  to  employ  even  weighting,  if  things  so  hap- 
pened that  the  uneven  weighting  really  involved  slumld  be  the 
correct  weighting,  then  our  operation  woidd  be  much  better  than 
what  we  intended.  IJut  this  would  liappen  only  on  the  very 
improbable  su])position  that  the  uuuibcrs  of  the  mass-units  of  the 
diiferent  classes  purchasal)le  at  the  first  period  were  directly  pro- 
portional to  the  sizes  of  the  classes  properly  estimated.  It  is  es- 
pecially unlikely  to  ha])]ien  because  the  custom  is  to  employ  larger 
units  for  the  more  iinpoi'taul  classes,  so  that  the  weighting  is 
more  apt  to  be  the  opj)osite  of  what  it  ought  to  be.  If  we  em- 
ployed the  same  mass-unit  throughout,  for  instance,  a  pound,  the 


I'OSSIHLK    ERRORS  165 

weig-htiug  would  be  iuversejy  according  to  the  preciousness ''  of 
the  chisses, — whicli  would  mostly  be  in  the  right  direction,  but 
enormously  exaggerated  in  some  instances,  and  generally  most 
irregular. 

This  em})loym(>nt  of  a  single  mass-unit  seems  to  have  the 
recommendation  that  it  does  away  with  the  multiplicity  of 
results  obtainable,  and  so  with  our  perplexity  about  deciding 
between  them.  But  it  is  not  so  ;  for  if  we  employed  a  weight- 
unit  throughout,  or  a  capacity-unit  throughout,  the  results  in  the 
averages  of  each  period  separately  and  in  comparison  would  be 
diiferent.  Evidently  neither  measure  is  more  authoritative  than 
the  other,  and  neither  provides  the  weighting  which  any  rational 
treatment  of  the  subject  would  recommend. 

It  is,  however,  questionable  whether  in  practice  any  of  these 
methods  is  worse  than  the  method  carrying  out  even  weighting 
— that  is,  whether  the  weighting  in  any  of  these  methods  de- 
parts more  from  the  true  weighting  than  does  even  weighting. 

§  4.  If,  then,  we  wish  to  employ  proper  weightmg,  and  think 
that  we  have  found  what  it  ought  to  be,  representing  the  weight 
for  [A]  by  a,  that  for  [B]  by  b,  that  for  [C]  by  C,  and  so  on  ; 
if  we  proceeded  as  before  and  introduced  this  weighting  for  the 
classes  in  the  averages  at  the  periods  separately,  and  then  com- 
pared them,  as  represented  in  this  formula 

31^,      a.{a,A)  +  h{b^B)  -h  c{c,C)  + 


3Irn       a(rr,.4)  +  \i{h^B)  +  c(ejC)  +  

and  then  dropped  tlie  A,  B,  C,  ,  wrongly  taking  them  for 

units,  or  as  merely  referring  to  the  classes,  and  so  used  the 
formula 

Mo.  _  aft^-Hb6^4-  CC3-I- 

Moi  ~  a  a^ -f- b  6^ -t- c  c^ -I- ' 


our  formula  would  still  be  wrong,  and  would  still  not  universally 
do  what  we  intended  ;  for  now  the  weights  of  the  variations 
would  be  (as  shown  in  Appendix  A,  I.  §  8),  not  a,  b,  C,  ,  as 

^  Preciousness  is  an  idea  in  economics  very  much  like  density  in  physics. 
Density  is  the  ratio  of  the  weight  of  a  body  to  its  volume.  Preciousness  is  the 
ratio  of  the  exchange-value  of  a  body  either  to  its  volume  or  to  its  weight. 


166  MATHEMATICAI,    FOinrrLATION 

intended,  hut  3l<i^,  b  6j,  C<\,    ,  that  is,  something  else,  irreg- 

uhir  and  haphazard,  containing  the  same  accidental  factors  as 
before,  namely,  the  quantities  at  the  first  period  purchasable 
with  one  money-nnit  and  measured  in  various  mass-units.  And 
there  M-ould  also  be  an  indefinite  number  of  results  obtainable, 
according  to  the  possible  combinations  of  possible  mass-units 
that  may  be  used.     But,  of  course,  if  the  reduction  had  been 

made  so  that  a^,  b^,  Cj,  are  units,  or  any  uniform  quantity, 

then  the  weighting  employed  would  be  the  one  intended  for  the 
variations.  Naturally,  with  the  proper  weighting  rightly  aj)- 
plied  there  can  be  only  one  result. 

Thus  in  our  previous  numerical  example,  sujjpose  at  each 
period  (or  on  the  whole)  we  found  the  class  wheat  to  be  twice 
as  important  as  the  class-  cotton  and  three  times  as  important  as 
the  class  coal,  and  wanted  to  use  this  weighting.  This  weighting 
for  the  variations  would  be  carried  out  in  the  following  manner, 

Mo.       1/3  2  5  \ 

indicating  a  rise  of  8|^  per  cent.  But  if  we  employed  the  same 
weighting  in  the  method  of  averaging  separately  on  the  unequal 
figures  above  instanced,  we  should  have  in  the  one  case 

Mq,  _  i(;3  X  3  +  2  X  2  -1-1  X  5)  _  9-^4-^5  _  18  _ 

^/oi  ~  K3  X  2  +  2  X  4  +  1  X  5)  -  6  -h  8  -h  5  ~  19  ~    '^  "^'' 

indicating  a  fall  of  o.;3  per  cent.  ;  and  in  the  other, 

Mq,      1(3  X  24  +  2  X  2  +  1  X  5)  _  72  +  4  +  :>  _  -^1  _        ,- 
J4,  "  K3  X  16  -h  2  X  4  +  1  X  5)  ~  48  -f-  8  -f-  5  "  01       '•'^-'' 

indicating  a  rise  of  32.7  ]xn'  cent.  These  (lis('r('|)aneies  are  due 
to  the  fact  that  in  the  lii'st  ojx'ration,  which  carries  out  the 
weighting  intended,  we  are  attaching  slightly  more  importance 
to  the  rise  of  wheat  than  to  the  fall  of  cotton  and  to  the  con- 
stancy of  coal  ;  in  the  second,  we  are  really  attaching  slightly 
more  importance  to  the  fall  of  cotton  (in  the  ])ro|)ortion  of  S  to 
6  and  5);  in  the  third  we  arc  really  attaching  much  more  im- 
portance than  we  intended  to  tiie  rise  of  wheat  (in  the  propor- 


1'()S.S1J{I>K    KIMtoltS  167 

tion  of  48  to  H  and  o).  AVhen  we  come  to  treat  of  the  measure- 
ment of  the  exchange-vahie  of  money  by  means  of  prices,  we 
shall  find  a  convenient  way  by  Avhich  the  right  result  may  in 
some  cases  be  very  conveniently  reached  by  the  use  of  this 
method  of  adding  before  dividing.  But  liere  no  such  method  is 
forthcoming.      For  the  operation  represented  in  tlie  formula 

J7^  _  K3xl|[2]  +2  X  i[4]  +1  X  1  [5] ) 
Mo.  -  K3  X   1  [2]  +  2  X  1  [4]  +  1  X  1  [5]) 

has  nothing  to  recommend  it,  as  its  luimerator  performs  the 
whole  operation  of  the  method  of  dividing  before  adding,  and 
its  denominator  is  pure  waste,  reducing  to  unity  and  not  affect- 
ing the  result. 

§  5.  Harmonic  averaginr/. — Intending  to  use  even  weighting, 
we  might  be  led  to  average  each  ])eri()d  and  to  compare  the 
averages  in  this  wav, 

1 

1  r   1         i         i 


31,,        n    [(a.^A)   '  (h,B)    '   (c^C) 


Mo,  ~ 1 

1    f     1  1  1 

+  77-T.r  +  ^-77^    + 

n 


{a,A)  ■   (6,7?)    '   (e,C) 

and  if,  as  before,  we  incorrectly  eliminated  A,  B,  C,  ,  we 

should  have,  after  reducing  and  transposing, 

1         1         1 


7W-       -  +  r  +  ~  + 

-M-02  «i  t>i  <^'i 


J/oi        111 

-  +  r+-  + 

«.,        0.,         t'2 

And  here,  too,  not  only  the  reasoning  by  which  this  result  is 
reached,  but  this  result  itself,  is  wrong.  For  (as  is  shown  in  Ap- 
pendix A,  III.  §  7)  this  is  the  formula  for  the  harmonic  aver- 
aging of  the  variations  with  weighting  of  —  for  [A] ,  of  7-  for 

[B] ,  of  —  for  [C] ,  and  so  on,  that  is,  with  weighting  inverse/}/ 
according  to  the  sizes  of  the  figures  at  the  first  period.      Hence 


168  MATHK>rATICAL    FORM  T  I.ATION 

a  multij)licity  of  results  would  be  obtaiued  according'  to  the 
mass-units  we  happen  to  employ.  Here  the  larger  the  mass-unit, 
the  smaller  the  number  of  it  purchasable  with  the  money-unit  at 
the  first  period,  and  the  greater  the  influence  of  the  variation ; 
and  the  smaller  the  mass-unit,  the  smaller  the  weight.  And 
here,  too,  it  is  conceivable  that  such  even  Aveighting  in  the  sepa- 
rate averages,  giving  uneven  weighting  in  the  average  of  the 
variations,  might  give  the  right  uneven  weighting,  although  this 
is  exceeding  unlikely  to  occur.  It  is  not  so  unlikely,  however, 
as  in  the  preceding  case ;  for  w^e  have  seen  that  the  weighting 
there  was  likely  to  be  the  opposite  of  what  it  ought  to  be. 
Here  the  Aveightiug,  being  the  inverse  of  that,  is  therefore  likely 
to  run  more  in  the  right  direction,  although,  of  course,  only  in 
an  irregular  way.  On  the  other  hand,  the  employment  here  of 
the  same  mass-unit  throughout  would  tend  to  make  the  error 
run  in  the  wrong  direction. 

If,  instead,  we  wished  to  employ  proper  weighting,  and,  finding 
it  the  same  at  both  i)cri(ids,  introduced  it  in  the  se]>aratc  aver- 
ages, and  compared  them  in  this  form, 


a  b  c 


Mo,       n'\{a.^A)      jb.^B)      {c,C) 


Mo, 


1  f    a  b  c 


and  in  the  same  manner  reduced  this  to 

a       b       c 

IT.,      a       b       c  ' 

«.,      o.,       c, 

we  should  again  be  performing  a  wrong  operation,  as  the  weight- 
ing in  this  formula,  for  the  variations  (as  shown  in  Ai)pen(lix  A, 

a    b    c 

III.  §  8)  is  not  a,  b,  C,  as  intended,  but  — ,  ^,  — ,  ,  and 

«j     0,      Cj 

the  correct  weighting  is  again  perverted  by  a  lia})hazard  factor. 


POSSIBLE    ERROns  169 

It  is  not  worth  wliilc  t(t  liive  numerical  examples  here,  or  to 
expatiate  further  on  the  errors  incurred  through  negligence  in 
the  treatment  of  harmonic  averages,  although  this  is  the  average 
wliieh  has  been  mostly  adopted  in  measuring  the  exchange-value 
of  money.  But  it  has  generally  been  adopted  from  the  opposite 
view-point  of  the  measurement  of  price-variations  by  means,  as 
we  shall  see  presently,  of  the  arithmetic  average,  in  treating  of 
Avhieh  we  shall  luu'e  to  revert  to  this  subject. 

§  ().  Geometric  averarjhig. — Here  everything  is  simpler.  The 
weighting  in  the  comparison  oi  the  averages  at  the  two  periods 
and  the  weighting  in  the  averaging  of  the  variations,  if  appar- 
entlv  the  same,  are  so  really  ;  for  it  does  not  now  matter  whether 
we  multiply  before  dividing  or  divide  before  multiplying.  If 
we  employ  even  weighting  in  the  one  case,  we  have  even  weight- 
ing in  the  other,  as  is  hereby  shown. 


Moo_v'a^-b^-c.^-- 

■■■:-{/ 

«i  ^1 

^2 

^hi      v^a^-b^-c^-  ■■ 

^1 

And  if  we  employ  the  same  uneven  weighting  in  each  separate 
average  and  in  the  average  of  the  variations,  we  really  have  the 
same  weighting  in  both  cases. 

The  wrong;  reasoning  bv  which  the  erroneous  results  were 
reached  in  the  first  two  cases  has  no  room  here.     In  reducing 


Mo.      v{a,A).{b,B).{c.fi) 


Mor      {>'{a,A).{b,B).{c^C). 

to  the  above  expressions  by  dropping  A,  B,  C,  from  the 

two  sides  of  the  fraction,  we  have  a  perfect  right  to  do  so  ;  for 
although  these  expressions  are  not  units,  they  are  the  same  on 
both  sides.  The  whole  expression  cannot  be  further  simplified 
by  dropping  the  radical  signs,  because  the  products  are  not  in 
the  same  ratio  as  their  roots. 

Now  the  same  apparent  weighting  being  the  same  real  weight- 
ing in  this  kind  of  averaging,  a  warning  is  still  needed.  Aver- 
aging the  periods  separately,  we  might  be  inclined  t<)  weight 
everv  figure  according  to  the  number  of  times  it  occurs,  what- 


170  .MA  rilKMATICAl,    FORM  T  LATK  )N 

ever  the  size  of  its  class  may  be  ;  and  if  these  numbers  happen 
to  be  the  same  at  both  periods,  the  comparison  of  the  averages 
would  be 

JIqo      '^0.2  •^2-^2  ^  ^  terms 

^^o\      'y  a^ .  6/ .  Cj* to  n  terms ' 

in  which  the  exponents  x,  y,  z,  refer  to  the  numbers  of  the 

mass-units  that  appear  in  trade,  and  n'  =  .v  -\-  y  -\-  x  -\-  io  n 

terms.  We  might  then  be  inclined  to  adopt  this  weighting  into 
the  method  of  averaging  the  variations, 


:!;='''(:;)■{«•(? 


^   '''  =  1/    (  -^^  )  .  {  ;i  )  .  (  -2  ) to  n  terms 

This  would  be  wrong.  The  results  would  be  various  according 
to  the  sizes  of  the  mass-units  accidentally  chosen.  For,  the 
larger  the  masS'Unit  chosen,  say,  for  [A] ,  the  smaller  would  be, 
not  only  a^  and  a.^,  (the  numbers  of  times  the  mass-unit  is  pur- 
chasable with  one  money-unit,  which  are  here  immaterial,  since 
their  j^roportion  is  always  the  same),  but  also  x  (the  number  of 
times  the  mass-unit  appears  in  trade),  that  is,  the  weight  of  this 
variation.  Therefore  this  weighting  would  be  haphazard.  If, 
in  order  to  avoid  the  multiplicity  of  results  which  would  ensue, 
we  adopted  only  one  mass-unit  throughout,  the  weighting  would 
be  influenced  by  the  relative  preciousness  of  the  articles  ;  and, 
as  before,  as  preciousness  may  be  measured  in  two  ways  (by 
c(miparison  of  the  exchange-value  either  Avitli  weight  or  with 
bulk),  there  would  still  be  two  distinct  acts  of  weighting.  The 
proper  Aveighting  is  that  which  we  have  already  discovered, — 
the  number  of  times,  not  the  accidentally  actual,  but  the  real  (or 
ideal)  economic  units  occur  in  each  class.  This  weighting  should 
be  employed  in  the  averaging  of  the  variations  ;  and  if  we  happen 
to  prefer,  for  any  reason,  the  other  method  of  averaging  each 
period  separately  and  comparing  their  results,  we  ought  to  em- 
ploy the  same  weighting  there. 

§  7.  One  more  point  deserves  to  be  considered  here.  We 
have  seen  that  the  errors  we  have  been  warning  against  we 
siiould  have  been  apt  to  slip  into  had  we  em[)loyed  the  method 


I'OSSIBT.K    KltUOHS  171 

of  notating  the  (luantities  at  the  second  period  independently, 
without  expressinti:  their  relation  to  the  quantities  at  the  first 
l)eriod.  We  should  have  been  especially  apt  to  fall  into  them 
had  we  additionally  employed  the  imperfect  method  first  noticed 
of  fornudating  exchange-value  relations  by  means  of  the  e({uiv- 
alences  of  mass-quantities.  For  then — we  may  here  confine 
our  attention  to  the  arithmetic  method — from  the  formula  for  the 
first  period, 

M^  ===  ftj A  <:-  6jB  =3=  CjC  === to  n  terms, 

we  might  have  deduced 

Mj  =c=-  («jA  +  ^,B  +  CjC  -I-  to  )i  terms) ; 

and  from  the  similar  formula  for  the  second  period  we  might 
have  deduced  a  similar  formula  of  the  combination,  and  then 
the  comparison  would  be  in  this  form, 

{a^A  -f-  62B  -I-  c^C  + to  n  terms) 


M, 


M   '  1 

^      -  (a,  A  -I-  6,B  +  e.C  + to  w  terms) 

in  which  as  the  terms  A,  B,  C,  merely  refer  to  mass-units 

of  the  diifereut  classes,  and  so  represent  units,  we  might — es- 
pecially if  only  one  mass-unit  had  been  employed  for  all  the 
classes — have  thought  ourselves  justified  in  omitting  them  and 
reducing  the  formula  to  this. 


M, 

«2  +  ^2  +  ^2  +   

ur 

a^  -\-  h^  +  e^  +  

— the  faulty  formula  already  treated  of.  Here,  too,  however, 
our  reasoning  would  have  been  w^ong,  as  is  obvious  if  A,  B, 
C,  referred  to  diiferent  mass-units.  But  even  if  they  re- 
ferred all  to  the  same  mass-unit,  our  reasoning  w^ould  have  been 
wrong,  as  we  are  not  dealing  with  cpiautities  of  weight  or  of  bulk, 
but  with  quantities  of  exchange-value  (namely  the  exchange- 
values  of  certain  weights  or  volumes  of  things).  Here,  for  our 
purpose,  a  ]iound  of  lead  is  very  diiferent  from  a  pound  of  feath- 


172  MATHEMATICAL    FolLMl'LATION 

ers.  It  is  only  a  dollar's  worth  of  lead  that  is  worth  as  much 
as  a  dollar's  worth  of  feathers. 

§  8.   This  kind  of  f  )rniulation  calls  for  some  remarks.     The 
arithmetic  formula 

M  -z^  -  (a  A.  -\-  f>B  -\-  cV  + to  n  terms) 

n 

may  be  read  :  "The  money-unit  (or  anything  else)  is  equiva- 
lent to  the  sum  of  the  quantities  of  the  things  it  can  exchange 
for  or  purchase,  divided  by  the  number  of  their  kinds";  or 
again  :  "The  purchasing  power  of  the  money-unit  (or  of  any- 
thing else)  is  the  power  of  purchasing  the  sum  of  the  quantities 
of  the  things  it  can  purchase,  divided  by  the  number  of  their 
kinds,"  the  supposition  being  that  the  kinds  or  classes  are 
equally  important.  And  similarly  if  we  used  this  formulation 
for  the  other  two  kinds  of  averages.  The  peculiarity  of  this 
formulation,  and  of  such  interpretations  of  it,  is  that  there  is  ref- 
erence only  to  the  mass-quantities  of  the  things  purchasable, 
and  no  reference  to  the  exchange-values  of  these  masses,  or  of 
money  in  their  classes.  Evidently  such  an  average  would  have 
manifold  results  accordino-  to  the  sizes  of  the  mass-units  em- 
ployed.  Hence  some  method  has  to  be  added  of  selecting  the 
mass-units.  One  is  to  use  always  the  same  mass-unit^ — either  a 
weight-unit,  or  a  capacity-unit.  With  the  addition  of  such  a 
method  (the  difference  between  the  two  not  generally  being  no- 
ticed), such  is  the  employment  often  made  of  the  term  "  purchas- 
ing power."  Yet  not  oidy  this  formulation  docs  not  yield  a 
mathematically  serviceable  formula,  but  also  these  interpretations 
of  it,  with  the  added  methods  of  selecting  the  mass-unit  just  no- 
ticed, and  the  term  "  general  purchasing  power  "  so  used  with 
reference  merely  to  the  mass-quantities  of  the  things  purchas- 
able, do  not  yield  clear  ideas.  The  things  purchasable  whose 
mass-quantities  alone  are  thus  united  and  divided,  or  averaged, 
form  only  a  ])romiscuous  conglomeration  of  variously  valuable 
things  with  many  and  great  qualitative  distinctions,  ^vhich  are 
entirely  ignored.  The  partial  similarity  in  this  method  of  meas- 
uring general  purchasing  power  with  the  true  method  of  meas- 


OF    PRICE    RKI.ATIONS  I  I  4 

uring  jwrticiilar  purcluisin^-  powers,  and  the  complete  orroneous- 
ness  of  it,  will  be  more  fully  j)ointe(l  out  in  a  later  Chapter. 

It  is  possible,  however,  to  put  a  true  interpretation  upon  the 
formulation  by  equivalence.  The  above-given  arithmetical  aver- 
age may  be  made  to  rejid  :  "  The  general  purchasing  power  of 
the  money-unit  (or  of  anything  else)  over  all  other  things  is 
equal  to  the  sum  of  its  particular  purchasing  powers  over  every- 
thing separately,  divided  by  their  number  " — under  the  sup])osi- 
tion  of  even  importance.  But  here  w^e  merely  have  exchange- 
value  under  another  name,  based  upon  a  formula  mathematically 
imperfect.  AVe  must  prefer,  therefore,  not  only  the  other  kind 
of  formulation,  but  also  the  term  "  exchange-value."  It  is  a 
canon  in  logic  that  we  should  avoid  the  use  of  two  terms  Avith 
identical  meanmg,  and  should  choose  the  term  which  more  clearly 
expresses  the  meanmg.* 

IV. 

§  1.  Thus  far  we  have  made  no  use  of  prices.  Yet  in  all 
monetary  matters  the  use  of  prices  is  a  great  help.  Almost  all 
the  attempted  measurements  of  the  exchange-value  of  money 
have  proceeded  by  drawing  averages  of  prices.  We  must  there- 
fore restate  the  previous  procedures  in  terms  of  prices. 

We  have  been  dealing  directly  with  the  exchange-value  of 
money  in  other  things.  Prices  .express  the  exchange-value  of 
the  other  things  in  money.  Therefore  the  variations  of  prices 
are  the  inverse  of  the  variations  of  the  exchange-values  of  money 
in  the  things  priced.  And  a  variation  of  an  average  of  the 
variations  of  prices  will  indicate  the  inverse  variation  of  an 
average  of  the  variations  of  the  exchange- values  of  money  in  the 
things  priced.     Hence  the  possibility  of  substituting  measure- 

■*  We  can  think  of  the  "  lifting  power  "  of  a  derrick  without  thinking  of  a  lift- 
ing power  in  the  things  it  lifts.  And  so  we  can  think  of  tlie  "  purchasing  power  " 
of  money  without  thinking  of  the  pui-chasing  power  of  the  things  it  purchases. 
In  fact,  this  term  "to  purchase"  alwaj's  means  that  we  give  money  for  some- 
thing. Hence  money  actually  has  purchasing  power,  in  this  proper  sense,  with- 
out anything  else  having  purchasing  power  ;  and  so  "purchasing  power  "  is  not 
a  correlative  term.  But  we  cannot  employ  the  term  "  exchange-value  "  without 
the  idea  of  exchange,  which  involves  correlative  exchanges,  and  consequently 
correlative  exchange-values. 


174  MATHEMATICAL  FORMULATIOX 

ments  of  prices  for  direct  measurements  of  the  exchauge-viilue 
of  money  in  other  things. 

If  we  start  with  mass-quantities  equivalent  to  the  money- 
unit,  and  with  prices  therefore  at  unity,  the  subsequent  variations 
of  the  exchange-values  of  the  money-unit  and  of  prices  will  be, 
as  we  have  seen  (in  Proposition  X.),  to  reciprocals  of  each  other. 
Now  M^e  commenced  our  previous  enquiries  always  by  supposing 

M  ^-  a  A  =0=  6B  -o  c  C  =0= ,  which  means  that  the  prices  of  a  A, 

of  6B,  of  cC, are  one  money-unit  each  (whence  the  price  of 

one  mass-unit  of  fAI,  nanielv  A,  is       of  B  ,,of  C-,andso 

■-    -"  'a  b  c 

on).  Therefore,  at  the  first  period,  if  we  draw  the  averages  of  the 
prices  of  these  equivalent  mass-quantities,  however  large  or  small 
these  be,  we  shall  always  get  unity,  as  before.  And  at  the  second 
period,  when  M  ==  a' a  A  -=  6'6  B  ===  c'c  C  === ,  the  prices  of 

a  A,  of  6  B,  of  c  C are     ,,77,     ,, Now  letting  P^  and 

P.,  represent  the  averages  of  the  prices  at  the  first  and  second 
periods  respectively,  as  Pj  in  every  case  is  1,  in  every  case 

P.  . 

-p^  =  P.„  that  is,  the  comparison  of  the  two  averages  is  the  same 

as  the  average  at  the  second  })eriod.  Therefore,  again  supposing 
that  we  are  dealing  with  classes  equally  important  at  each 
period,  or  over  both  the  periods  together,  by  inverting  the  order 
and  placing  the  harmonic  average  first  and  the  arithmetic  second, 
for  a  reason  which  will  appear  immediately,  we  have 

P„  1  1 


P~l/lll  \1  ' 

b'       g'  J 


p;=n  («'+//+.'+ >  (1^'-') 

P^  =  l/i  .1.1  =  ^  (12,  3) 

^1  a     t)     (  y  a'   b'   a' 


OF    PRICE    KEI.ATIONS  175 

But  from  the  oxchano^c-valuc  comparisons  wo  liad  (4,  i  :i) 

§"=^«'+A'+r'   +  ), 

Mo, 1 

Mo,~  1  /  1       1    .   I 


n  \a' 


+  5'+?  + 


Mqi  n/     ,      1  ,        I  _ 

•  Wor^        • 

Thus  it  is  evident  that  in   the  order  given  the  first  fornuila   for 

P 

:p^  vields  a  result  the  reciprocal  of  the  result  of  the  first  formula 

J/  >  P 

for  —""  J  the  second  formula  for  ^5^  a  result  the  reciprocal  of  the 

31  o 
result  of  the  second  formula  for  ,  ,-  ,  and  the  third  formula  for 

-"^01 
P  M ., 

^  a  result  the  reciprocal  likewise  of  the  third  formula  for  ^j^" . 
Pi  -^'^  01 

Now  the  first  formula  here  expresses  the  harmonic  average  of 
the  prices  at  the  second  peoiod  (or  of  the  variations  of  the  prices), 
while  the  first  formula  there  expressed  the  arithmetic  aver- 
age of  the  variations  of  the  money-unit's  exchange-values  ; 
the  second  formula  here  the  arithmetic  average,  the  second  there 
the  harmonic  ;  but  the  third  here  and  the  third  there  both  ex- 
press the  geometric  average.  Thus  when  the  harmonic  average 
of  prices  shows  a  rise  of  the  exchange-values  of  other  things  in 
money  by  p  per  cent,  to  1  -|-  p  times  their  former  level,  which 
means  a  fall  of  the  exchange-value  of  money   in   the  things 

1  1  p 

priced  to  :, by  1  —  :, or per  cent.,  this  fall  is  in- 

^  1  -{-p    -^  1  -\-p        1  +p 

dicated  by  the  arithmetic  average  of  the  exchange-values  of  the 
money-unit  in  those  other  things  ;  and  reversely  if  the  former 
shows  a  fall.  When  the  arithmetic  average  of  prices  shows  such 
a  rise  of  the  exchange-values  of  other  things  in  money,  which 
means  such  a  fall  of  the  exchange-value  of  money  in  the  other 
things,  this  fall  is  indicated  by  the  harmonic  average  of  the  ex- 
change-values of  the  money-unit ;  and   revcn-scly  if  the  former 


176  MATHEMATICAL  FORMULATION 

shows  a  fall.  But  when  the  geometric  average  of  prices  shows 
such  a  rise  of  the  ex  change- values  of  other  things  in  money,  which 
means  such  a  fall  of  the  exchange- value  of  money  in  other  things, 
this  fall  is  indicated  also  by  the  geometric  average  of  the  ex- 
change-values of  the  money  luiit  ;  and  reversely  if  the  former 
shows  a  rise.  Again,  if  on  one  set  of  variations  the  Jiarmotiio 
average  of  prices  shows  constancy  of  the  average  of  the  exchange- 
values  of  other  things  in  money,  which  means  also  constancy  in 
the  average  of  the  exchange-values  of  money  in  other  things, 
this  result  is,  on  the  same  variations,  indicated  by  the  arithmetic 
average  of  the  exchange-values  of  the  money-unit.  If  on  an- 
other set  of  variations,  the  arithmetic  average  of  prices  shows  such 
constancy,  it  is  indicated,  on  this  set  of  variations,  by  the  Jiar- 
monic  average  of  the  exchange-values  of  the  money-unit.  But 
if,  on  still  another  set  of  variations,  the  geometric  average  of 
prices  shows  such  constancy,  it  is  indicated,  on  the  same  set, 
likewise  by  the  geometriG  average  of  the  exchange-values  of  the 
money-unit. 

This  relationship  between  these  averages  is  in  accordance  with 
a  well-known  mathematical  theorem  concerning  these  averages. 
The  harmonic  average  of  any  given  quantities  is  the  reciprocal 
of  the  arithmetic  average  of  the  reciprocals  of  those  quantities. 
Reversely  the  arithmetic  average  of  any  given  quantities  is  the 
reciprocal  of  the  harmonic  average  of  the  reciprocals  of  those 
quantities.  But  the  geometric  average  of  any  given  quantities 
is  the  reciprocal  of  the  geometric  average  of  their  reciprocals.  It 
is  precisely  with  such  quantities  and  their  reciprocals  that  we 
are  dealing. 

We  can  therefore  find  the  average  variation  of  the  exchange- 
values  of  money  in  all  other  things  by  drawing  the  average 
variation  of  the  prices  of  all  things,  but  noticing  that  the  har- 
monic averages  of  prices  is  the  one  which  gives  the  inverse  of 
the  arithmetic  average  of  the  exchange-values  of  money,  the 
arithmetic  average  of  prices  the  one  whicli  gives  the  inverse  of 
the  harmonic  average  of  the  exchange-values,  while  the  geo- 
metric average  of  prices  corresponds  also  with  the  geometric 
average  of  the  exchange- values. 


OF    PraC'E    T! FIXATIONS  177 

The  j)r()l)l(Mn  still  remains  as  t(»  which — il"  any — of"  these 
averages  is  the  right  one.  The  ])roj)er  order  of  the  en([uirv 
would  .seem  to  be  as  to  which  is  the  right  average  to  express  the 
variation  of  the  excliange-values  of  the  money-unit  in  all  other 
things.  If  we  find  this  to  be  the  arithmetic  average,  we  shall 
know  that  the  proper  average  to  draw  of  prices  is  the  harmonic. 
If  the  right  average  for  the  exchange-values  turns  out  to  be  the 
harmonic,  then,  and  only  then,  will  the  arithmetic  average  of 
prices  be  the  right  one.  But  if  the  right  average  in  the  one 
case  is  found  to  be  the  geometric,  the  geometric  will  also  be  the 
right  average  in  the  other.  It  Avould  seem  that  to  turn  the 
problem  around  and  to  start  with,  or  to  confine  ourselves  to,  an 
enquiry  concerning  the  proper  averages  of  jirices  Avould  give  us 
a  wrong  point  of  view  and  put  us  at  a  disadvantage. 

§  2.  The  formulae  for  the  averages  of  prices,  of  course,  are 
not  to  be  confined  to  the  form  in  A\'hich  the  prices  at  the  second 
period  are  stated  as  reciprocals  of  the  variations  of  the  exchange- 
values  of  the  money-unit  in  the  things  jjriced,  that  is,  in  the 

fractional  forms  — ,  r^,  — , We  may  give  them  upright 

integral  forms.      Let  us  employ  the  accented  Greek  letters,  a', 

/3',  y' ,  ,  to  represent  the  prices  of  a  A,  6  B,  eC,  at  the 

second  period — these  being  the  quantities  which  at  the  first 
period  were  all  priced  at  1.00,  so  that  their  prices  at  the  second 
period  directly  express  the  variations  of  the  prices.     Then,  as 

1,1           1 
o.'  rei)laces  — ,  ^i'  j-,,  y'  —,,  in   the  preceding  formulse,  we 

have  the  followinu' : 


1       1/1         1         1  \' 


(13,0 


p'=iK  +  /'^'  +  r'  + ),  (13,0 


P 

12 


^^  =  ^V-,9'r' (13,3) 


17s  MATHEMATICAL    FORMILATIOX 

We  cau  also  empl(jy  the  other  method  of  uotatiou.      We  may 

represent  the  prices  of  anii  mass-uuits  of  the  various  classes  at 

the  first  period  by  «„  y9j,  )\,  ,  and  the  prices  of  the  same 

quantities  of  the  same  classes  at  the  second  period  by  «.„  fi.^,  j.^, 

Then  the  variation  in  the  preceding  formuhe  rejn'esented 

«., 
by  a'  is  here  represented  by     ",  and  that  represented  there  by 


q 
/9'  is  here  represented  by    ^,  and  so  on.     And  the  preceding 

formulse  become  the  following  : 

F 
P 


P,  1 


^'    V  «2  i^2  To  J 

P,^i/s     I     r.     \  (14^,) 

^  =  i/^.^.n (14,3) 

In  these  formuhe  the  terms  are  the  reciprocals  of  the  terms  in 

formnlffi  5,  i-s,  that  is,  —  =  —,—"=,   ,   -  =  -^ ,  and  so   on,  and 
'  «,       a.,   ji,       b.,    r,       (-2 

P. 

inversely.     Hence  each  formula  here  expresses  for  p"therecip- 

3f , 
rocal  of  the  formula  there  given,  in  the  same  order,  for  ^~  • 

It  is,  of  course,  more  natural  to  use  the  prices  of  the  custo- 
marily employed  mass-units  of  the  different  kinds  of  goods  ;  and 
it  is  more  convenient.  In  practice  all  reductions  of  prices  at  the 
first  period  are  labor  wasted. 

V. 

§  1.  The  mere  introduction  of  uneven  weighting  into  these 
formula?  is  likewise  easy.     Tiie  weighting  is,  of  course,  the  same 

*  These  three  forimilie,  confined  to  two  terms,  were  first  given  by  Jevons,  B. 
23,  p.  121,  and  then,  in  tlie  more  general  form,  by  Walras,  B.  09,  pp.  12-13,  B.  70, 
p.  432. 


OF    IMUCK    liKl.ATIOXS  179 

as  in  the  measurement  of  tlie  exchan^t»- values  «»i'  money  in  the 
other  classes.  The  wei<>;htin»j:;  of  the  arithmetic  averajre  of  the 
variations  there  will  come  into  the  harmonic  average  of  the 
variations  here  as  it  there  entered  the  harmonic  average  ;  and 
reversely.  The  formuhe,  then,  when  we  are  justified  in  using, 
or  when  we  do  use,  single  weighting,  will  be 


Pj       1  /a       b       c 

n 


/a       b       c  \' 

I  -/  +^:57  +  P  +  to  n  terms! 


P,  H 


2  =  -^^  (a«'  -f  b;^'  +  Cj-'  +  to  n  terms),         (15,  2) 


in  which  n"  =  a  +  b  +  C  +  to  n  terms  ;  or  again 

-  =  ^ ^ ^,  (1  <J,  • ) 


-   (  a^  +  b(^  +  C-^i  +  to  n  terms  ) 

i"V   «2       /^2       r.  / 

— -  =  —  (  a  — -  +  b '— -  +  C  ^-  +  to  n  terms  ),  (1 G,  2) 

Pi       "V     «i  /^.  "A  / 

The  same  comments  are  suggested  by  these  fornudse  as  by  those 
for  the  exchange- values  of  money.      It  is  indifferent  how  large 

or  small,  integral  or  fractional,  the  numbers   a,  b,  C,  be, 

so  long  as  they  are  in  the  proper  proportions  to  one  another.  And 
a  change  in  the  weighting  without  a  variation  in  the  prices  (or 
with  a  uniform  variation  of  all  prices)  produces  no  change  in 
the  result  (in  agreement  with  Propositions  XLIV.  and  XLV.). 

1  These  foruiuhe  for  the  unevenly  weighted  averages  have  rarely  been  stated. 
Westergaard  gives  the  arithmetic  formula  in  a  cumbrous  form  (see  Appendix  C, 
III.  ?  1)  ;  and  adds  that  a  "similar  alteration  "  may  be  given  to  the  other  simple 
averages,  but  he  does  not  give  it,  and  the  expression  "similar"  is  misleading. 
They  are  unnoticed  by  all  the  other  writers  cited  in  the  Bibliography  except 
Edgeworth,  who  gives  the  third  in  its  logarithmic  form,  as  noticed  below  in  NoteH. 


180  :\iaiiii;ma  ricAi,   i'(»i;mi  i.aiion 

AVe  may  here  add   that  these  and  all  the  preeediiio-  proper  Ibr- 
mulse  also  carry  out  Proposition  XXXVI. ^ 

With  high  numbers  for  the  weights  the  third  of  these  formulse 
might  seem  unworkable.  Indeed  it  would  be  unworkable  even 
with  small  numbers  for  the  weights,  or  with  all  the  weights 
alike, — since  their  sum,  when  many  classes  are  dealt  with,  would 
involve  the  extraction  of  a  large  root, — but  for  the  help  which 
may  be  rendered  by  logarithms.  With  the  aid  of  logarithms 
the  third  formula  is  no  longer  difficult  to  execute,  even  with 
large,  or  with  minutely  fractional,  figures  for  the  weights.  This 
use  of  logarithms  was  made  by  Jevons,  although  he  employed 
only  even  weighting.  With  uneven  weighting,  the  logarithm 
of  the  result  may  be  obtained  in  several  ways,  among  which  the 
following  may  be  noted  : 

(1)  log"  p'  =  ^^f,  (a  ■  leg-'  +  b  •  log'y'  +  to  n  termsj, 

P,  1 

(-)  l*'g  p"  ~  ^^1  {(^'l'*??  "-y  +  b-log  /5.,  + to    /(   terms) — 

(a  •  log  «j  +  b  •  log  /9^  + to  n  terms)!, 

P  1 

(.3)  log     -  =     „  {a  (log  a^  -  log  «,)  +  b  (log  ,i -  log  ;9,)  + 

to  ;;.  terms}. 

The  first  is  the  o])eration  we  should  naturally  adopt  if  we  had 

already  reduced  the  prices  at  the  first  period  to  1.00  (or  to  100), 

a., 

—  then   being  re])laced    by   u!   already  ascertained,  and  so  on. 

The  second  would   be  the  simplest,  if  the  weighting  were  even, 
the  expressions  a,  b,  C, then  dropping  out.'^     The  third   is 

2  This  and  a~remark  in  Appendix  A,  I.  ^10  end,  deserve  attention.  They 
show  that  wlien  we  know  approximately  the  variation  of  tlie  average,  it  is 
more  important  tliat  we  should  seek  to  be  aecurate  with  the  weighting  of  the 
classes  whose  variations  are  greatly  divergent  from  that  of  the  average  than  with 
the  weighting  of  the  classes  whose  variations  are  nearly  the  same  as  that  of  the 
average.    See  also  Note  21  in  Chapter  IV.,  Sec.  V.,  §  10. 

'  Very  nearly  in  this  form,  with  even  weighting,  the  formula  is  given  by 
Walras,  B.  01,  p.  7.  A  formula  very  nearly  in  the  first  form,  with  uneven  weight- 
ing,  has  since  been  given  by  I'llgeworth,  B.  59,  p.  287. 


OF    IMMCK    KKLATIOXS  181 

generally  the  most  usi^fiil  operation,  savinu-  the  divisions  needed 
in  the  first  and  the  donbling  <>f  the  multiplieations  in  theseeond.' 
§  2.  We  may,  of  course,  here  as  Avell  as  in  tlie  direct  meas- 
urement of  the  exchange-value  of  money,  need  to  use  double 
weighting.  The  formuUe  embodying  double  weighting  in  the 
case  of  prices  may  be  directly  derived  from  the  formulte  pre- 
viously given  for  double  weighting  in  the  other  case  (formulae 
11,  is).  We  must  still  employ  some  expressions — .r„  x.^,  y^,  y.,, 
etc, — to  represent  the  numbers  that  are  produced  or  consumed 
at  each  period  of  the  mass-units  whose  prices  are  given. 
And  we  must  remember  that  wo  must  make  two  inversions,  first 

p         ]\f(ji 
of  the  numerators  and  denominators,  for  ~    =  —    ,  and  second 

of   the    exchange- value    symbols    and    the    price    symbols,    for 

-^  =  — '  and  reverselv,  and  so  on.     Then  those  formuUe  become 

these,  in  the  same  order  : 

T>       ^(--^'^+-  + ton  terms) 

P,  -   1  /.r„      y.,       z.,  V  ^     '   ^ 

^        ^    -^  +  ^  +    -  + to  a  terms  ) 

p        ^^  (•*^2«2  +  ^2/^2  +  ^2/'2  + to  >*  terms) 

-^  =  h ,   (1 7,  2) 

'       -^  ,  G^\«i  +  yA  +  ■iu^  + to  "  terms) 


P        w     '2^^ '  ^2' '  ^2"^ '  ^  *^  terms 


2 


1/ 


^1      ?5'^«i''i  •  /^i^i  ■  Ti'-  to  yi  terms 


(l',^') 


The  anticipatory  remark  may  here  be  made  that  in  the  future 
course  of  these  pages  we  shall  find  no  use  for  any  of  these 
formulae  except  the  second,  which  is  the  general  formula  for 
double  weighting  in  the  arithmetic  average  of  prices.  This 
second  formula  may  be  expanded  in  full  as  follows, 

*  Jevons  used  three-place  tables.  It  would  be  advisable,  however,  to  use  tables 
with  not  less  than  six  places. 


182                              ^[ATIIEMATICAL    FORMULATION 
'g2«2  +  VA  +  Va  +  

P2  «2  +  2/2  +  '"^o  +   


Pi      •'«i«i  +  2/1/^1  +  ^iTx  + 


(17,^-0 


•'»^i  +  2/1  +  «i  + 
which  reduces  to 


Pi      -^^1+2/^  +  21^  + :r3  +  ^,+  2,.+ '^     ^-'  ^ 

this  last  being-  the  best  form  for  it. 

Here,  as  l)efore,  it  is  evident  that  in  all  these  formuhe  the  re- 
sults will  be  variously  affected  by  the  sizes  of  the  mass-units  we 
happen  to  choose.  Here  also,  therefore,  a  method  of  selecting 
the  mass-units  must  be  established.  And  this  method  need  not, 
or  it  may  be,  indicated  in  the  formula  itself.  It  is  not  indicated 
in  the  formula  used  by  Drobisch,  who  therefore  employed  the 
simple  formula  here  given,  after  first  telling  how  the  mass-units 
are  to  be  chosen.  It  is  indicated  in  the  formula  used  by  Pro- 
fessor Lehr,  which  therefore  becomes  more  complex,  though  the 
real  difference  between  his  method  and  Drobisch's  is  not  in  the 
formula,  but  in  the  method  of  choosing  the  mass-units,  and  his 
method  is  not  so  difficult  of  application  as  Drobisch's.  Both 
these  methods  use  the  second  formula,  which  admits  of  other 
variations  through  the  employment  of  several  different  methods 
of  selecting  the  mass-units. 

In  this  second  formula  there  is  the  peculiarity  that  the  sizes 
of  the  mass-units  have  no  effect  upon  the  terms  x^a^,  x./z.^,  etc., 
that  is,  upon  any  of  the  terms  in  the  first  half  of  the  formula 
in  the  form  last  given.  For  the  larger  the  mass-units,  the  larger 
will  be  a  in  the  same  pro])ortion,  and  x  will  be  smaller  in  the 
inverse  proportion,  so  that  their  product,  xa,  will  be  miaffected, 
and  so  in  all  the  other  classes,  and  at  both  periods.^     Hence  in 

^K.g.,  if  [B]  be  wheat  and  y^  represent  a  thousand  quarters,  tlien  /3j  represents 
the  price  of  a  quarter — let  us  say  eight  dolhirs,  wiierefore  y^  fti  =  1000  <  8  =  8000. 
If  instead  we  used  l)ushels,  y^  would  represent  eight  thousand  busliels,  but  ^i 
would  represent  one  dollar,  so  that  we  should  still  have  yi  fi^  =  8000  X  1  =  8Q00. — 
This  peculiarity  exists  in  the  corresponding  second  formula  for  the  exchange- 
value  of  money  ( 11, 2),  expressing  the  harmonic  average  ;  for  there  no  influence  is 
exerted  upon  the  terms  ,  ^,  etc.,  since  the  larger  is  y^,  tlie  larger  also  is  61  in  the 

same  proportion,  and  the  ratio  remains  the  same.     Thus  in  the  above  example 
1000  ^  8000 
1/8  ~     1    * 


OF    I'lMCK    UKLATIONS  ]  S.'J 

this  Ibnn  of  this  furinuUi  the  first  halt'  rcmuins  always  the  same, 
and  the  modifications,  if  made  at  all,  are  to  be  made  only  in  the 
second  half. 

This  formula  may  be  simplified.  In  the  first  half  the  whole 
denominator  is  merely  the  total  valuation  of  all  the  goods  pro- 
duced and  consumed  in  the  first  period  at  the  prices  of  the  first 
period  ;  and  the  whole  numerator  is  a  similar  total  valuation  for 
the  second  period.  Naturally  these  parts  of  the  formula  remain 
unchanged  whatever  be  the  mass-units  employed.  In  the  second 
half  the  whole  numerator  is  the  simple  inventory  of  all  the  mass- 
units  of  the  goods  produced  or  consumed  in  the  first  period  ;  and 
the  whole  denominator  is  the  similar  inventory  for  the  second 
])eriod.  Naturally  these  parts  are  aifected  by  the  sizes  of  the 
mass-units  in  which  these  numbers  are  reckoned.  Now  the  parts 
in  the  first  half  represent  the  total  money-values  of  the  goods  at 
the  first  and  second  periods,  and  hence,  as  wholes,  may  be  repre- 
sented by  Vj  and  V., ;  and  the  i)arts  in  the  second  half  repre- 
sent the  total  mass-quantities  of  the  goods  at  the  two  periods 
respectively,  and  hence,  as  wholes,  may  be  represented  by  Qj 
and  Q^.     Then 

is  a  simplified  form  of  the  second  formula. 

The  real  nature  of  this  second  formula  is  indicated  in  the 
second  form  in  which  it  has  above  been  put,  which  may  be  sim- 
plified into  this. 

For  here  it  is  seen  that  the  formula  is  merely  that  of  a  compari- 
son between  an  average  ])rice  of  all  goods  at  each  period.  For 
at  each  j>eriod  the  total  money-value  of  all  goods  collectively  is 
divided  by  the  total  quantity  of  all  the  goods  ;  which  is  the 
plain  method  of  (arithmetically)  averaging  the  prices  of  all  goods 
at  each  period.      But  the  total  "  quantity  "  of  all   the  goods  is 


184  MATHEMATIf'AI.    FOHMl'LATION 

the  total  sum  (»f  the  nuniber.<  of  the  mass-units  of  all  the  goods, 
wliich  sum  is  aifected  bv  the  sizes  of  the  mass-units  chosen  to 
measure  the  goods  bv,  so  that  the  averages  are  arbitrary  until 
the  proper  method  of  selecting  the  mass-units  is  found.  This 
fact  of  its  representing  a  comparison  between  two  ordinary 
(arithmetic)  averages  is  what  gives  this  second  formula  its  su- 
perior recommendation  over  the  other  two. 

It  might  be  thought  that  there  would  be  a  similar  recom- 
mendation for  the  first  formula  here,  because  it  corresponds  to 
the  first  formula  for  the  exchange-value  of  money,  in  which  the 
separate  averages  are  the  arithmetic.  But  in  (arithmetically) 
averaging  at  each  period  the  exchange-value  of  money  in  all 
goods  by  dividing  the  sum  of  its  exchange-values  in  them  by 
fJieir  quantity  there  does  not  seem  to  be  so  nuich  sense  as  in 
(arithmetically)  averaging  at  each  period  the  exchange-value  of 
all  g(»ods  in  money  by  dividing  the  sum  of  their  exchange-values 
in  money  by  their  quantity.  There  is  congruity  between  the 
money- values  of  connnodities  and  the  quantity  of  commodities  : 
there  is  none  between  the  conmiodity-values  of  money  and  the 
(juantitv  of  commodities. 

VI. 

i;  1.  The  averaging  of  prices  gives  occasion  for  similar  mis- 
carriages of  intention  as  the  averaging  of  the  exchange- values  of 
money.  Jevons's  assertion  that  there  is  no  average  of  prices, 
and  consequently  no  variation  of  the  average  of  prices,  must  be 
regarded  as  erroneous,  except  in  the  sense  that  there  is  no  real 
or  absolute  average  already  given  to  us  in  nature.  We  must 
(.'onstruct  the  average,  and  can  do  so  wrongly  or  rightly  ;  where- 
fore the  problem  really  is  to  iind  how  to  construct  it  rightly. 
In  the  attempt  to  solve  this  problem  two  methods  have  presented 
themselves.  The  one  is  to  construct  separate  averages,  one  at 
each  period,  on  the  mass-quantities  that  exist  at  each  period, 
variouslv  reckoned  according  to  the  mass-units  chosen  ;  where- 
upon the  comparison  lies  between  these  two  averages.  The  other 
is  likewise  to  construct  separate  averages,  and  to  compare  them, 
but  to  use  in  each  the  same  weighting — as  if  the  same   mass- 


Ki;i:()i;s   ixciiutKi)  1  (ST) 

quiintitics  existed  at  both  the  jK'riods  compared.  Now  wheu- 
ever  this  second  luetliod  is  employed,  it  is  the  same  as  some 
method  of  averaging"  the  price  variations  with  single  Aveighting ; 
and  reversely,  whenever  single  weighting  is  employed  in  aver- 
aging price  variations, — or  whenever  the  method  of  averaging 
the  price  variations  is  em})loyed,  since  in  this  there  can  only  be 
single  weighting, — it  is  the  same  as  some  method  of  averaging 
the  prices  at  each  period  sei)arately  and  comparing  the  averages. 
Therefore  the  single  weighting  for  the  price  A'ariations  which  is 
involved  in  the  method  of  separately  averaging  the  prices  and 
of  comparing  the  averages  ought  to  be  the  same  as  woidd  l)e  the 
weighting  did  we  avowedly  ]Hirsue  the  method  of  averaging  the 
price  variations ;  for  it  Ayoiild  be  absurd  to  employ  a  method  of 
comparing  aA'crages  of  prices  that  is  really  the  same  as  a  method 
of  averaging  the  price  variations,  and  to  employ  in  it  a  system 
of  weighting  for  the  price  variations  M'hich  Ave  would  reject  if 
Me  consciously  employed  the  method  of  averaging  the  price 
variations.  Thus,  as  before  (Sect.  III.  §  2),  we  have  a  criterion 
that  the  method  of  compdrhu/  separate  price  averages  with,  the 
same  weighting  in  each  should  always  agree  with  tJie  method  of 
averaging  the  price  variations  loith  the  proper  loeighting  for  these, 
whatever  this  weighting  ma}'  be  ;  for  at  all  events  it  is  often  easy 
to  reject  flagrantly  improper  weighting,  or  weighting  that  has 
no  reason  in  its  favor. 

This  criterion  is  of  importance  because  the  single  weighting 
for  averaging  the  price  variations  involved  in  comparing  the 
separately  drawn  averages  is  not  always  the  same  as  the  weight- 
ing used  in  each  of  these  averages.  It  is  always  the  same  if  we 
construct  at  the  tirst  period  rather  a  level  of  prices,  which  breaks 
up  into  an  uneven  range  of  prices  at  the  second  period  ;  for  in 
this  case  to  draw  an  average  of  these  later  prices  is  really  to 
draw  an  average  of  the  variations  of  the  prices,  "with  the  same 
weighting.  But  if  we  start  at  the  first  period  with  various 
prices,  there  is  generally,  though  not  always,  a  divergence  be- 
tween the  weighting  in  the  separate  averages  and  the  Ayeighting 
for  the  price  variations  involved,  and  to  the  ignorant  hidden,  in 
the  comparison  of  those.      It   is  here  that  the  opportunity  for 


186  MATHEMATICAL    FOHMUI.ATIOX 

error  enters  in  this  method  of  coiiipariiiir  averages,  systems  of 
weighting  for  the  price  variations  lieing  admitted  which  wonld 
never  have  been  allowed  had  their  existence  been  perceived.  To 
the  errors  so  occasioned  we  may  for  the  present  mostly  confine 
our  attention,  but  also  noticing  an  allied  error  in  a  method  using 
double  weighting.  Again  the  averages  need  t(t  be  treated  se])a- 
rately. 

§  2.  Haiitionie  averaging  of  prices: — This  corresponds  to  the 
arithmetic  averaging  of  the  exchange-values  of  money,  but  its 
form  is  the  same  as  the  harmonic  averaging  of  tliose  exchange- 
values.  The  possible  error  here  is  that  of  supposing  we  are 
using  even  weighting  in  using  tlie  following  fonnula, 

1         1         1 

P.  _  «i      /^i      Yi 

P ,  "111 

''■2  /^2  l'-2 

or  that  we  are  using  weigh'tmg  according  to  a,  b,  C,  in  using 

the  following, 

a       b       c 

P,      a       b      c 

'  +    3    +-   +  

«2  P2  Yl 

— that  is,  in  comparing  the  harmonic  averages  of  anij  prices 
(the  prices  of  any  mass-units)  at  eacli  period  separately,  in  each 
of  these  averages  the  same  even  or  uneven  weighting  being 
used.      For  in   the  first  of  tliese  fbrnud:e  the  weighting  of  tlie 

price  variations  is  reallv  according  to  -^,  — ,  — ,  ,  and  in  the 

'h   /^i    Yi 

1 •    •            T           a     b     c  ,  .        .... 

second  it   is  accorduig  to  — ,   — ,  — ,  ,  and   \\\    neitliei"  is  the 


o. 


/'i     /i 


1     /'I     I  I 


weigliting  what  is   intended  unless  a,  =  .9,  =  j',  =  ,  that   is, 

unless  all   ])ri(;es  have  been   reduced   to   the  same  figure  at  the 

first   period.'      We   know   thai   ^/,  =      ,  .9,  =  ,,;',  =     ,  and   so 

on.      Thus  the  weighting  here  is,  in  the  first  fonnula,  according 
1  Kc'c  Appi'iidix  A,  III.  iJi^  7  iiiul  S,  iilready  referred  to. 


KlfKons    INCl  Ix'UKI)  187 

to  <7,,  6,,  Cj, ,  and  in  the  second  acoordinij;'  to  arr,,  b/>,,  Ce^, ; 

which  is  the  same  wei*;litintj;  as  we  discovered  in  the  correspond- 
iuo^  forms  of  the  arithmetic  averaging  of  the  exchange-values 
of  money  (above  in  Section  III.  §§  ^^  and  4). 

Various  complications  could  be  added  which  would  occur  in 
comparisons  of  the  results  obtained  for  subsequent  periods  by 
comparing  every  one  of  these  with  tlie  same  basic  period.  But 
as  this  method  has  rarely  been  used  Av^e  need  not  tarry  over  it. 

§  3.  Ai'ithiiu'fie  arenujinx/  of  prices. — This  corresponds  to  the 
harmonic  averaging  of  the  exchange- values  of  money,  and  its 
errors,  while  the  same  in  form  as  those  which  may  occur  in 
arithmetically  averaging  the  exchange-values,  are  really  the 
same  as  those  which  may  occur  in  harmonically  averaging  them. 
As  these  latter  have  only  been  lightly  treated,  and  as  this  is  the 
method  which  has  mostly  been  used,  more  attention  is  to  be  paid 
to  it  here. 

In  the  comparison  of  the  price  \ariations  we  can  start  with 
the  prices  of  any  mass-units.  Hence  the  simplest  form  of  error, 
and  the  one  earliest  to  appear,  was  to  think  we  may  simply  aver- 
age the  prices  of  any  mass-units  of  commodities  at  the  two 
periods  and  compare  them.  The  operation  is  according  to  this 
formula, 

J,         \«.  +  il  +  /'.  +  ) 

^_  »       

^'      ^K +  /^i  +  n  + )' 

which  reduces  to 

P,  _  «.  +  1%  +  r,  4- 

Pi     «i  + /3^ -h  r, +  ••••■• ' 

so  that  this  method  consists  merely  in  comparing  the  sums  in 
lists  of  prices  at  two  or  more  periods. 

This  form  is  wrong  because,  as  a  method  of  averaging  the 
variations  of  prices  it  contains  weighting  according  to  the  sizes 
of  the  figures  in  the  denominator — the  prices  at  the  first  period.^ 
But  as  these  prices  have  been  hit  upon  without  reference  to  the 

^  See  Appendix  A,  II.  ^  7,  already  referrcil  to. 


188  MATHEMATTCAT.    FORMri-ATIoX 

sizes  of  the  elasses,  the  Aveightiug  is  haphazard — as  ah-eady  no- 
ticed ill  the  preceding:  Chajiter.  As  the  prices  are  larger  the 
larger  the  mass-units  chosen,  the  weighting  will  be  larger  the 
larger  the  mass-nnits.  This  agrees  with  what  we  have  found  in 
the  case  of  the  inipro])er  method  of  comparing  harmonic  aver- 
ages of  the  exchange-values  of  money.  Now  if  it  be  claimed 
that  all  we  want  is  to  compare  averages  of  prices,  and  not  to 
average  variations,  the  error  of  this  position  is  shown  by  the 
fact  that  this  method  would  give  as  many  different  results  as 
there  are  combinations  of  mass-units  that  might  be  employed. 
Among  these  the  right  result  would  be  hit  upon,  or  ap])roachcd, 
only  if  the  prices  at  the  first  period  happened  to  be  directly  ac- 
cording to  the  sizes  of  the  classes  rightly  determined. 

This  haphazard  method,  which  may  be  named,  after  the  writer 
Avho  first  employed  it,  Dutot's  method  (see  Appendix  C,  L),  has 
rarely  l)een  em})l()yed  completely.  Yet  it  lias  frequently  been 
in\olved  in  another  method  which  was  specially  invented  to 
avoid  it,  and  which  maybe  called  Carli's  method  (see  Appendix 
(",  II.).  This  other  method  is  to  average  variations  with  even 
weighting  by  reducing  all  the  prices  at  the  first  period  to  a  uni- 
form level ;  and  then, — in  a  variation  introduced  by  Evelyn, — 
in  order  to  avoid  the  trouble  of  repeating  this  operation  for  every 
coni[»arison,  the  first  ])criod  is  used  as  a  common  basis  with  which 
everv  subsequent  j)cri()(l  is  directly  compared.  This  method  car- 
ries out  its  intention  \vhen  conn)aring  each  subsecpient  p(>riod  with 
the  basic  period  ;  but  whenever  the  result  obtained  fi)r  one  sub- 
sequent period  is  compared  with  the  result  obtained  fi)r  another, 
the  comparison  acquires  an  entirely  different  character.  The 
comparison  of  the  second  period,  for  instance,  with  the  first, 
whose  prices  have  been  reduced  to  unity,  is  in  accordance  with 
this  formul;i, 

p;=i^  =  ^«  +  /V  +  r;  + ), 

and  the  coin|)arison  of  the  third  with  the  same  first  is 

P  =l^.  =  ;^«+/V+r:/  + ); 

1 

whci'cforc  the  comparison  of  these  two  is  as  follows, 


Enrvons  txcttiiut)  189 


Pa 

+ 

^' 

+ 

hi 

+ 

■■) 

P, 

n  ^  - 

+ 

/^/ 

+ 

1  -1 

+ 

■■■) 

diicli 

may  alsi 

II  he  written 

thus, 

P 
P 

3  

+ 

J 1 

«., 

+ 

f^. 

+  " 

+ 


'^-1        /^i         /I 

Here  the  avera^inu,-  of  the  price  variations  is  no  Ioniser  with 
even  weigliting-  :  it  is  with  uneven  weighting  according  to  the 
variations  of  the  prices  at  the  earlier  of  the  two  ])eriods 
compared.  Consequently  the  result  obtained  in  this  way  is  not 
the  same  as  would  be  the  result  if  we  compared  the  third  ])eriod 
directly  with  the  second,  employing  even  weighting  ;  and  the 
result  we  do  get  depends  upon  the  accident  as  to  what  period  we 
happen  to  start  with  as  the  base,  and  what  price  variations  have 
taken  place  since.  Of  coui'se  tlie  price  variations  have  no  re- 
gard for  the  sizes  of  the  classes,  so  that  the  uneven  weighting 
here  is  purely  fortuitous.  To  be  sure,  by  the  time  of  the  second 
period,  the  variations  may  not  be  large,  so  that  the  Aveightiug 
will  not  much  depart  from  the  even  weighting  intended.  But 
the  further  we  go  from  the  original  period,  the  more  and  more 
freakish  becomes  the  w^eighting.  This  method  has  now  been 
employed  by  the  Econoinid  in  a  series  covering  fifty  years.  It 
is  amusing  to  think  what  queer  weighting  may  be  involved  in 
comparing  the  fiftieth  with  the  forty-ninth  year.  Yet,  after  all, 
this  weighting  may  not  be  more  incorrect  than  the  even  weight- 
ing itself. 

That  the  comparison  of  the  averages  of  prices  using  (tnt/  prices, 
with  a])parent  even  weighting,  may  give  an  indefinite  number 
of  different  answers  is,  of  course,  simply  due  to  the  fact  that  an 
indefinite  number  of  different  weightings  may  be  used,  according 
to  the  prices  that  happen  to  exist  and  to  be  hit  upon  at  the 
first  period.  Yet  this  fact  of  the  A^ariability  of  the  results  ob- 
tainable has  been  urged  as  a  reason  fi)r  rejecting  the  system  of 


190  MATHEMATICAI.    FORMULATIOX 

index-numbers  altogctlicr.  AVriters  liave  toyed  with  xarious 
prices  that  might  be  used  at  the  first  period,  and  on  the  same 
price  variations  have  naturally  obtained  dilfereut  results.  Then, 
not  perceiving  the  reason  for  the  ditfereuee  in  the  results,  they 
have  concluded  that  a  method  giving  such, a  variety  of  answers 
can  have  no  validity.'  Of  course  this  criticism,  based  upon  ig- 
norance, and  merely  ])laying  off  one  incorrect  form  against  an- 
other, has  no  validity  against  the  correct  form,  whatever  this 
may  turn  out  to  be  when  it  is  discovered. 

Now  the  variability  of  the  results,  it  might  be  thought,  could 
be  obviated  by  confining  our  comparisons  always  t<^  the  prices 
of  the  same  mass-unit.  If  we  did  so  with  the  intention  of  em- 
ploying even  weighting, — in  fact,  employing  even  weighting  in 
the  averages  of  prices  at  each  period  separately, — there  would 
still  be  uneven  weighting  when  we  view  the  result  as  an  aver- 
age of  the  price  variations.  For  the  prices  would  be  large 
or  small  directly  according  to  the  preciousness  of  the  classes, 
and  so  the  result  would  be  the  same  as  if  we  averaged  the 
variations  using  weighting  directly  according  to  the  precious- 
ness of  the  classes  at  the  first  period — an  absurd  system  of 
weighting,  as  the  importance  of  the  classes  is  very  different 
from  their  preciousness,  and,  in  fact,  apt  to  be  the  opposite. 
Moreover  this  method  would  not  obviate  the  difficulty,  as  there 
would  be  two  results  obtainable  according  as  we  used  through- 
out the  same  weight-unit  or  the  same  capacity-unit.  Another 
objection,  but  of  minor  im])ortance,  is  that  this  method  could 
not  ])ossibly  be  applied  to  land,  and  with  difficulty  to  gases. 
That  it  could  not  be  apj)lied  to  labor,  however,  would  be  in  its 
favor.  We  need  not  dwell  longer  on  this  method,  since  it  has 
never  been  employed.  But  the  notice  of  it  is  justified  because 
its  principal  feature,  the  use  of  the  same  mass-unit,  has  been  in- 
corporated in  another  meth<»d,  to  be  examined  presently. 

§  4.  Another  way  in  which  variability  of  the  answers  may  be 
avoided   is  by  taking    ac('(tunt    of  the   mass-quantities  of  the 

^  So  Pierson  in  the  pajxT  referred  to  in  a  note  to  B.  122,  and  C.  W.  Oker,  The 
fallacy  of  iiuUx-numbers,  .Journal  of  Political  Eeononiy,  (^liieaRO,  Vol.  IV.,  1S96, 
pp.  r)]7-r»l!). — The  proper  answer  has  reeently  been  fjiven  hy  Padari,  H.  I  11,  i)p. 
ltlH-l<t!t. 


ERRORS    INCURRED  191 

classes  produced  and  consumed  at  one  of  the  ])eri()ds,  or  of  some 
average  mass-quantities,  treating  them  as  if  thev  were  the  same 
at  each  period.  This  method  has  not  infre({uently  been  em- 
ployed, and  may  be  called  Scrope's  method,  after  the  person  who 
first  suggested  it  (see  Appendix  C,  IV.).  Let  x  represent  the 
number  of  the  mass-units  of  the  class  [A]  found  to  be  pnjduced 
or  consumed  by  the  comnumity  at  large  during  a  given  period 
or  periods,  and  y  the  number  of  the  mass-units  of  the  class  [B] 
similarly  found,  and  so  on.  Then  at  the  first  period  the  com- 
munity can  buy  this  total  mass  of  goods  by  expending  the  fol- 
lowing sum  of  money  :  xa^  -f-  7//?^  +  z-j\  -\-  to  n  terms, — the 

prices  «„  /5j,  j-j, of  course  being  the  prices  of  the  same  mass- 
units  of  [A],  [B],  [C], whose  numbers  are  represented  by 

X,  y,  z, And  at  the  second  period  the  same  total  mass  of 

goods  can  be  purchased  for  this  sum  of  money  :  xa.^  -f-  ^/9.,  +  zy., 

-f to  n  terms.      If  we  divide  each  of  these  sums   of  money 

by  the  number  of  mass-units  purchased — namely  by  x  -\-  y  -\-  z 

-\- to  n  terms,  which  we  may  represent  by  n' , — the  quotient 

may  be  taken  to  be  the  average  price  of  a  mass-unit  (an  average 
mass-unit,  so  to  speak)  of  the  goods  ;  and  therefore  a  variation 
of  these  averages  may  be  represented  thus, 

P   •  -,  (^^2-^y^2  +  ^r2+ ) 

which  reduces  to 

^2  ^  ^^«2  +  y/^2  + '^ra  + ^ 

Pi      •^•«i  +  //y^i  +  ~-/'i  + ' 

Or  we  might  have  started  sim])ly  with  this  last  formula  as  the 
expression  of  the  variation  of  the  total  prices  of  the  same  total 
quantity  of  goods.  Therefore  a  method  doing  this,  although  it 
does  not  necessarily  involve  an  arithmetic  averaging  of  the  prices 
at  each  period,  is  the  same  as  a  method  which  does  make  such 
arithmetic  averaging. 

There  is  little  to  recommend  this  method  regarded  as  a  com- 
parison of  averages;  for  the  averages  are   average  prices  of  a 


192  MATIfEMATICAI.    I'OKMII.ATIOX 

rather  nondescript  mass-unit  ofnoods.  I'ut  tliese  averau'es  have 
the  merit  that,  vary  the  mass-units  enqdoyi'd  as  wc  may,  the 
two  averati'es  vary  in  the  same  ])r(ti)(>rti<)n,  so  that  the  result  of 
the  eomparison  of  them  is  uuatfeeted  ;  wh(>ref  )re  also  it  is  the 
same  as  that  ofa  mere  comparison  of  the  total  sums  without  any 
averaging  at  all.  *  Noav  viewed  in  this  second  form  there  is 
much  to  reeonnnend  this  method,  provided  the  mass-quantities 
produced  or  consumed  of  every  class  are  constant  over  both  the 
periods  compared.  For  then  we  are  really  coiuj)arino-,  so  to 
speak,  the  total  price  at  each  period  of  the  same  mass  of  goods 
actually  ])roduced  or  consumed  at  each  period. 

In  this  case  we  know,  l)y  our  mathematical  analysis,  ^  that 
while  in  each  of  the  sepai-ate  averages  of  the  prices  the  weights 

of  the  classes  are  x,  y,  z, ,  that  is,  the  numbers  of  times  the 

mass-units  chosen  are  produced  or  consumed,  yet  in  this  method 
viewed  as  a  method  of  arithmetically  averaging  the  |)rice  varia- 
tions the   weighting  is   according  to  xa^,  yfi^,  -jj-,, ,   that  is, 

according  to  the  total  exchange- values  of  the  classes  at  the  first 
period,  or,  more  simply,  it  is  what  w^e  have  calknl  the  ^veight- 
ing  of  the  first  period.  This  we  have  already  denounced  in  the 
])receding  diapter  as  an  absurd  system  of  weighting.  AVe  shall, 
however,  also  find  that  tliis  method  may  be  viewed  as  a  method 
using  other  kinds  of  averaging  of  the  price  variations,  with  other 
weighting,  and  so  avoids  the  absurdity  of  using  the  weighting 
only  of  the  first  period.  But  we  must  postpone  further  exami- 
nation of  this  method,  api)lied  to  this  case,  till  a  later  Cliapter. 
Here  what  needs  to  be  noticed,  is  that  in  Scrope's  method,  in  all 
its  forms,  regarded  as  a  method  of  arithmetically  averaging  price 
variations,  the  weighting  is  according  to  x(/.^,  yfi^,  zy^, 

"^  It  is  phuii  that  all  the  terms  used—///?!,  yfi^,  2}'i,  etc.— aro  tiie  same  wliole 
quantities  whatever  be  the  sizes  of  the  mass-units  chosen.  Cf.  above  Sect.  V., 
Note  5.  Hence  it  is  indifferent  what  mass-units  be  employed.  This  is  shown  at 
length  by  Padan,  B.  141,  pp.  19r)-198.  Yet  some  writers  employing  this  method 
liave  thought  it  necessary  to  state  that  we  ought  always  to  employ  tlie  same  mass- 
unit.  So  Laspeyres,  B.  26,  p.  'Ml,  and  Paasche,  B.  .S3,  p.  171.  (Also  Lehr  has 
made  a  similar  mistake  with  regard  to  his  method,  as  will  be  pointed  out  on  a 
later  occasion.)  In  this  method  the  variability  of  answers  has  already  been 
avoided  by  use  of  the  total  money-values  and  elimination  of  the  bare  mass- 
quantities,  and  so  does  not  need  addition  ni'  tlir  ollici-  iiutliod  of  avoiding  varia- 
l)ility  by  the  ns(>  ofa  single  mass-unit. 

■'■'Sec  Appendix  A,  II.  ^7    already  referred  to. 


Tf  the  mass-([u:uititi('s  of  any  of  the  classes  vary  from  the  one 
period  to  tlie  other,  provided  they  do  not  all  vary  alike,  tlie 
above  recommendation  for  this  method  vanishes.  For  this  case 
several  variations  upon  this  methcjd  have  been  suggested.  Some 
writers  have  recommended  using  only  the  mass-(]uantities  of  the 
first  period,  others  the  mass-quantities  only  of  the  second  [)eriod,, 
and  others  a  mean  between  these  two,  as  we  have  seen  in  the 
preceding  C-hapter.  Furthermore,  almost  all  the  economists  and 
statisticians  who  have  advised  the  adoption  of  this  method,  or 
have  actually  adopted  it,  in  any  of  its  varieties,  have  had  the 
habit,  similar  to  the  treatment  of  Carli's  method,  already  noticed, 
of  stringing  out  the  comparisons  over  many  periods  in  a  series, 
comparing  each  of  the  later  periods  directly  with  the  first  as  a  basic 
period.  Some  have  used  always  the  same  mass-quantities,  either 
those  of  the  first  period,  or  of  some  other  one  period,  or  a  general 
average,  in  all  these  comparisons.  Now  if  the  mass-quantities 
of  the  first  or  of  some  other  one  period  be  used,  the  comparisons 
between  the  later  periods  really  are  averages  of  the  price-vari- 
ations Avith  weighting  according  to  the  money-values  of  the 
mass-quantities  of  the  given  period  at  the  prices  of  the  less 
removed  of  the  two  periods  compared  (or  of  the  further  removed, 
if  the  two  periods  be  prior  to  the  basic  period), — which  is  a 
wholly  anomalous  and  senseless  system  of  weighting,  the  weight- 
ing also  being  different  in  every  comparison,  though  this  Avas 
not  intended.  If  a  single  general  average  mass-quantity  of 
every  class  be  used,  the  later  comparisons  are  equally  anomalous 
in  form,  but  have  the  merit  of  not  altogether  disregarding  the 
mass-quantities  of  the  periods  compared.  There  is  little  to 
recommend  this  method  in  theory,  though  it  may  perhaps  turn 
out  to  yield  answers  near  to  the  trutli  in  practice.  If,  lastly^ 
the  mass-quantities  of  the  second,  or  later  period,  be  used  in  all 
the  comparisons  with  the  first  or  basic  period,  a  peculiar  com- 
plication arises.  For  now,  when  we  compare  two  later  periods 
with  each  other,  we  are  employing  the  mass-quantities  of  both 
these  periods — that  is,  we  are  employing  double  weighting,  and 
double  weighting  without  any  authenticated  method  of  selecting 
the  mass-units,  hence  an  altogether  al)surd  method,  stnu'k  upon. 
13 


}U4  MATnF:.MATICAL    FOKMrLATroN 

hy  cliancc  incivly.  'I'liis  nu'tliod,  witlidiit  Iviiowlc'do-c  of  its  eoii- 
."sc'cjueiices,  was  actually  reconinu'iided  by  Paasche,  aud  may  be 
called  Paasclie's  variety  of  Sca"(»]X''s  method.''  The  consequence 
Avhich  it  brin<>s  about  was  evidently  never  intended  ;  for  if  it 
were,  the  same  method  of  usinsi,-  the  weighting  of  both  periods 
ought  to  be  eni])loyed  in  comparing  the  subse(pu'nt  periods  with 
the  first,  and  if  this  double  weighting  is  not  wanted  in  those 
comparisons,  it  is  not  wanted  in  the  (tthers.  In  fact,  Paasche 
himself  objected  to  the  use  of  double  weighting,  and  criticised 
Drobisch  for  using  it." 

A  similar  inconsistency  exists  in  another  method.  This  is  in 
the  similar  serial  form  of  the  method  of  arithmetically  averag- 
ing the  ])rice  variations  with  uneven  Mcighting — the  method 
iirst  employed  by  Arthur  Yoiuig.  For  if  this  method  be  em- 
])loyed  in  such  a  series  always  with  the  ^veightiug  of  the  later 
period  in  every  comparison  of  a  later  period  with  the  common 
first  or  basic  period, — as  it  has  been  employed  by  Mr.  Palgrave, 
wherefore  this  may  be  called  Palgrave's  variety  of  Young's 
method, — then  in  the  inverse  of  every  comparison  of  jirices  be- 
tween any  of  the  later  periods  (or  in  the  direct  comparison  of  the 
exchange-values  of  money)  there  is  use  of  the  weighting  of  each 
of  the  periods  compared.^  Here  also  the  use  of  double  weighting 
Nvas  never  inteiuU'd,  and  the  same  inconsistency  exists  ;  for  if 
the  comj)arisons  between  the  later  periods  are  correct,  the  com- 
|)arisons  of  the  later  periods  with  the  first  period  cannot  be  cor- 
rect, and  then  correct  comparisons  would  be  dependent  upon  in- 
correct comj)arisons, — which  is  absurd. 

§  5.  This  unperceived  merging  of  some  of  the  common 
methods  of  employing  single  weighting  into  methods  employing 
<louble  weighting,  suggests  that  avc  should  liere  examine  the 
methods  avowedly  employing  double  weighting  (with  the  arith- 
metic averaging  of  prices),  doing  so  in  the  comparison  of  the 
second  period  with  the  first  as  well  as  in  every  other  comparison. 
But  we  may  confine  our  attention  for  the  present  to  the  first  of 
these  methods  ever  invented,  namely  to  Drobisch 's,  examining 

*  For  its  formula  see  Appendix  (",  IV.  {S  2  (2),  and  Cor  tiie  naturu  of  its  mass- 
units,  Appendix  C,  V.  ^  4. 

*  B.  33,  pp.  172-173. 

*  For  the  formula  see  A|)p(iMli\  ( ',  III.  i^4. 


i:i;i;()i;s   in<tti»t!i;i)  195 

this  because  it  also  involves  another  of  the  features  above  no- 
ticed, namely  the  use  of  a  sintj^le  mass-unit  in  all  the  classes  for 
the  sake  of  avoiding-  variability  in  the  results.  The  mass-unit 
preferred  by  Drobisch  was  a  wei*>;ht-unit — a  hundred-weight, 
though  the  si/e  of  the  common  weioht  is  indifterent.  He  wanted 
the  finding  at  each  period  of  the  average  price  of  the  Aveight-unit 
of  all  goods,  the  average  being  the  arithmetic  with  weighting 
according  to  the  numbers  of  the  weight-units  in  every  class  at 
each  period.  He  claimed  that  by  so  doing  we  should  get  a  sort 
of  absolute  average  })rice,  and  so  disprove  Jevons's  denial  of  an 
average  price. ^ 

The  formula  for  this  method  is  the  general  formula  for  the 
arithmetic  average  of  prices  with  double  weighting  above  given 
(17,  :!,  3),  the  method  of  selecting  the  mass-unit  being  presup- 
posed. The  method  itself  may  be  illustrated  by  the  folloAving 
example.'"  If  in  a  country  during  a  certain  first  period  say  ten 
millions  of  the  same  w' eight-units  of  goods  (this  figure  being  ob- 
tained by  adding  up  all  the  numbers  of  weight-units  of  all  the 

classes — .i\  -\-  //,  -|-  z^  -\-  to  n^  terms  =  10)  are  produced  and 

consumed  at  a  total  valuation  of  fifteen  million  money-units 
(this  sum  being  obtained  by  adding  up  the  numbers  of  weight- 
units  of  every  class  multijilied  by  their  prices — x^a^  -|-  i/^ft^  -)-  zj^ 

4-  to  >/,  terms  =  ]')),  there  is  an  average  price,  represented 

by  -i-|-  or   Ih  money-units,  for  the  weight-unit.      Then  if  at  the 

second  jteriod  twelve  million  weight-units  (,t^  +  .^2  "f"  '"^•'  +  

to  n.,  terms  =  12)  are  produced  and  consumed  at  a  valuation  of 

twenty -one  million   money-units  (x.//..^  -\-  y.^S.^  -f-  z.^y.^  -\-  to  n.^ 

terms  =  21),  the  average  of  prices,  or  the  average  price  of  the 
mass-unit,  would  be  ^|  or  If  money-units.  Therefore,  he  would 
conclude,  not  merely  an  average  of  prices,  but  simply  the  price 
of  the  luass-unit  of  goods,  has  risen  from  1|  to  1|,  or  as  from 
6  to  7,  that  is,  by  16f  per  cent.,  and  inversely  money  has  de- 
preciated as  from  7  to  0,  by  14.29  j)er  cent. 

In  this  method  the  use,  not  of  the  arithmetic  average,  nor  of 

double  weighting,  but  of  a  common  mass-unit  may  be  shown  to 

be  wrong. 

9  B.  29,  pp.  44-45,  B.  30,  p.  153. 
*"  See  also  Appendix  C,  Y.  iJ  1. 


196  MA  rirK>rATrcAL  foumila  riox 

Tlie  use  tit'a  coiniiKin  niass-imit  lucaiis  that  tlic  axcraiic  di-awn 
at  each  period  is  an  average  price  of  an  average  mass,  so  to 
speak,  of  all  goods,  or  rather,  it  is  sinij)ly  the  price  of  a  small 
mass  made  up  of  small  fractions  of  all  goods,  tlicsc  fractions 
being  proportioned  to  one  anotiier,  by  weight  or  by  bulk,  accord- 
mg  as  are  the  weights  or  the  bulks  of  the  classes  relatively  to 
one  another.  This  small  compoimd  mass,  or  AnaxagoT'ian 
homoeomeria,  is  a  sort  of  representative  mass-imit,  being  com- 
posed in  the  same  way  as  the  whole  mass  of  goods.  It  may  be 
differently  composed  at  each  |)erio(l,  and  its  jiricc  may  be  differ- 
ent at  each  period,  Drol)iseh  claimed  that  the  variation  of  its 
price  represents  the  variation  in  the  money-value  of  all  goods, 
and  inversely  the  variation  of  the  exchange-value  of  money  in 
all  goods  (other  than  money).  This  claim  is  unfounded,  and 
violates  several  j)rinciples. 

In  the  first  place  no  reason  is  offered  why  the  use  of  a  com- 
mon weight-unit  was  chosen  rather  than  a  common  ca})acity- 
unit.  Yet  two  differently  varying  prices  would  be  obtained 
acc(n'ding  as  the  one  or  the  other  is  used  of  these  kinds  of  mass- 
units.  Thus  this  method  does  not  succeed,  excej)!  arbitrarily, 
even  in  the  attempt  to  avoid  the  fault  of  admitting  variability 
of  ecjually  good  and  hence  nuitually  destructive  results,  and  suc- 
cumbs to  a  criticism  already  advanced  against  several  other 
suggested  or  possible  methods. 

In  the  second  place,  because  of  this  feature  in  it,  tlu-  follow- 
ing absurdities  can  be  ])roved  of  this  method  :  (1)  betwctii  two 
jK-riods  between  whicli  n(»  variation  of  any  pi'ice  whatsoever 
takes  place,  if  any  irregular  change  takes  })lace  in  the  mass- 
quantities,  the  result  will  indicate  a  variation  of  prices  "  (con- 
trary to  Propositions  XXVII.  and  XLIY.) ;  (2)  between  two 
periods  between  which  all  pri<'es  ha\"e  varied  in  exactly  the  same 
]>ro])ortion,  if  any  irregular  change  taUes  place  in  the  mass- 
(piantities,  the  result  will  indicate  a  variation  ditferent  fi'om 
that  common  \ariation  (contrarv  to  Propositions  XVII.  and 
XIjV.)  ;  (-J)  between  tw(»  pei-iods  between  which  all  jirices  have 

11  Thin  is  tlic  criticisni  made  l)y  Laspeyres,  B.  2(i,  p.  ;i(is.  Drobisfh  replied  by 
reafliriiiiiif,'  ids  posii ion,  asseverating  that  tins  is  as  it  should  l)e,  H.  HI,  pji.  42")- 
42(5. 


i:i;i;()i;s   iN('ri!in;i)  197 

risen  somewhat,  if  (•crtaiii  changes  take  phice  in  the  niass-(i[nan- 
tities,  it  is  possible  tliat  the  resnlt  may  even  indicate  a  fall  of  the 
average  price ;  and  conversely.'"  These  absnrdities  are  not 
dne  to  the  nse  of  donble  weighting;,  bnt  to  the  nse  of  double 
weighting-  along  ^vith  this  method  of  selecting  a  common  mass- 
unit.  They  are  to  l>c  found  in  some  other  methods  using  double 
A\'eighting,  where  they  are  due  to  its  conjunction  with  some  other 
(consequently)  improper  method  of  selecting  the  mass-units. 
Thus  all  three  are  found,  upon  inspection  of  the  formulae,  in 
Paasche's  variety  of  Scrope's  method  and  in  Palgrave's  variety 
of  Young's  method,  so  far  as  these  use  double  weighting,  that  is, 
in  comparisons  between  later  periods,  in  a  series  founded  on  one 
j)eriod  as  a  base.  In  Lelir's  method,  rather  curiously,  the  second 
and  third  absurdities  are  to  be  found,  but  not  the  first.  Again 
all  three  are  found  in  the  last  two  forms  into  which  a  rather 
vague  method  invented  by  Professor  Nicholson  may  be  analyzed.'^ 
But  a  method  using  double  weighting  with  the  arithmetic  average 
may  be  discovered  that  is  free  from  all  these  absurdities. 

In  Drobisch's  method  it  is  not  difficult  to  see  why  double 
weightiuii'  with  use  of  a  common  mass-unit  leads  to  these  ab- 
surdities.  All  prices  remaining  constant,  an  increase  in  the 
quantity  of  ex[)ensive  goods,  the  prices  of  which  are  above  the 
average  at  the  first  period,  tends  to  raise  the  average  at  the 
second  period  (and  reversely  a  decrease) ;  but  an  increase  in 
the  quantity  of  cheap  articles,  the  prices  of  which  are  below  the 
average  at  the  first  period,  tends  to  lower  the  average  at  the 
second  (and  reversely  a  decrease) ;  and  the  average  will  be 
changed  according  as  the  one  or  the  other  of  these  movements 

^^  E.  g.,  in  the  above  numerii'al  example,  the  following  variations  would  be 
possible.     Suppose  at  the  first  period  things  were  thus  : 

cheap  goods  8  mill,  ewts.,  value  8  mill,  dollars  ;  price  |  =  1.00, 
costly      "       2     "         "  "     7     "  "  "     5  =-3.50, 

total  1(1     "         "  "    15     "  "  "    lg  =  1.50; 

and  at  the  second  thus  : 

cheap  goods  7     "  "  "     (J     "  "  "6=    .86, 

costly      "      5      "  "  "    15     "  "  "    i,i  =  3.00, 

total  12     "  "  "   21      "  "  "   ti=  1-75. 

Here  the  whole  collection  of  cheap  goods  has  fallen  in  price,  and  also  the  whole 
collection  of  costly  goods,  wherefore  it  is  possible  that  every  article  has  fallen  in 
price.    Yet  the  average  of  the  whole  has  risen  in  price. 
^^  8ee  Appendix  (',  V.  i;  .'i. 


198  MATIIFMATICAI.    F(  )I:M  T  I.A  TION 

predomiuatos.  And  all  prices  rising,  or  their  average  variation 
being  a  rise,  the  former  change  of  quantities  Avill  raise  the  aver- 
age still  higher,  and  the  latter  will  })revent  it  from  rising  so  high 
— perhaps  will  Icavt'  it  at  constancy,  or  even  occasion  a  fall ;  and 
reversely  if  all  prices  fall,  or  their  average  variation  is  a  fall.'* 
It  is  evident  that  the  disappearance  of  a  cheap  article  and  the 
appearance  of  a  dear  one  have  the  same  effect  as  the  variation 
of  a  cheap  article  into  a  dear  one  ;  and  reversely  the  disappear- 
ance of  a  dear  article  and  the  appearance  of  a  clieap  one.  A 
single  disappearance,  or  a  single  ap])earance,  has  half  this  influ- 
ence. Thus,  prices  remainmg  constant,  or  varying  in  a  given 
degree,  whether  this  method  shall  indicate  a  rise  or  a  fall  of 
prices,  or  a  greater  or  a  smaller  rise  or  fall,  will  depend  upon 
whether  the  change  in  the  weighting  happens  to  fidl  more  on 
expensive  goods  or  more  on  cheap  goods. '^ 

Thirdly,  even  at  each  of  the  periods  separately  the  weighting 
employed  by  Drobisch  is  wrong  ;  for  his  weighting  is  according 
to  the  mass-quantities,  ^vhich  weighting  we  have  seen  to  be  im- 
proper (in  Chapter  IV.,  Sect.  III.  §  3).  A  reason  wliy  it  is 
incorrect  is  that  the  mass-quantities  used  have  different  precious- 
uess,  which  also  deserves  to  be  counted,  or  otlierwisc  the  height 
of  the  average  will  depend  uj)on  the  accident  whether  dear  or 
cheap  goods  haj)pen  to  abound.  Still,  if  we  tried  to  correct 
this  error  by  weighting  the  mass-units  at  each  period  according 
to  their  preciousness,  as  well  as  according  to  their  masses,  we 
should  get  no  determinate  result,  because  at  each  period  the 
average  would  return  t(t  the  lowest  ])rice  we  liappcned  to  take 
as  our  basis  (probably  a  mass-unit  of  the  poorest  (piality).  The 
two  averages  would  have  no  connection  with  each  other,  and 
the  comparison  of  them  would  have  no  meaning.  To  get  this 
connection  we  need  to  employ  a  mass-unit   in   every  class  that 

i*  Thus  in  the  example  in  Note  12  the  supposed  increase  of  tiie  falling  goods 
was  in  the  expensive  goods  and  the  decrease  in  the  cheap  goods,  and  the  result 
was  a  rise. — The  general  niatheniatical  principles  are  given  in  Appendix  A,  VIT. 
(^  4,  with  reference  back  to  I.  jSjS  8-10. 

^^'  Hence  Drohisch's  method  can  he  true  only  if  the  niass-(iuaiitities  do  not 
vary,  or  all  vary  in  the  some  jiroportion.  This  was  perceived  t)y  J^ehr,  who  like- 
wise condemns  Drobisch's  method  for  using  such  economically  dissimilar  mix- 
tures of  things,  B.  (iS,  pp.  41-42.  The  last  criticism  is  also  made  liy  Zuckcrkandl, 
B.  115,  p.  247,  B.  IK),  p.  245. 


i;i;i;()i;s   iNcritiiKi)  109 

has  the  same  exchangc-vahie  over  l)(»th  the  j)eri()ds  together  as 
in  every  other  elass.  This  is  the  ini})roveinent  iutroduced  by 
Professor  Lehr,  wliose  method,  liowever,  as  we  shall  find  in  a. 
later  Chapter,  does  not  correctly  carry  out  the  improvement  in- 
tended, wherefore  the  true  method  is  still  something  else. 

Lastly,  a  general  reason  why  Drobisch's  method  is  wrong  is 
that  it  has  been  invented  without  any  regard  for  the  principles 
of  simple  mensuration.  We  have  seen  in  Chapter  III.  that  we 
must  compare  variations  only  in  two  similar  worlds.  Neither 
of  the  methods  there  pointed  out  for  obtaining  such  similarity 
are  here  observed.  The  same  troul)le  exists  in  the  varieties  of 
Scrope's  and  of  Young's  methods  introduced  by  Paasche  and 
by  Palgrave.  In  comparing  a  later  period  with  a  first  period 
on  a  given  weighting,  we  are  comparing  variations  in  two  sim- 
ilar worlds,  which  so  far  is  correct;  and  in  comparing  another 
later  period  with  the  first  on  another  weighting,  we  are  compar- 
ing variations  in  another  set  of  similar  worlds  ;  but  in  compar- 
ing the  results  of  these  two  measurements  we  are  comparing 
things  in  two  diiferent  worlds. 

It  should  be  noticed  that  in  many — in  fact,  in  most — subjects 
of  statistics  we  do  wish  to  use  double  weighting  with  the  ([uan- 
tities  as  they  happen  actually  to  be  reported,  that  is,  to  com- 
pare averages  at  diiferent  periods  l)uilt  on  varying  quantities  of 
things  whose  individuality  is  reckoned  without  regard  to  the 
attribute  we  are  measuring.  Here  the  above  principles  of 
sim])le  mensuration  are  not  violated,  for  the  good  reason  that 
these  are  not  subjects  of  simple  mensuration.  And  if  we  are 
comparing  things  in  dissimilar  worlds,  the  very  point  is  that 
we  are  dealing  with  dissimilar  worlds  and  want  to  compare 
them.  Thus  to  measure  the  measure  of  length  at  dififerent 
periods,  we  must  deal  with  similar  worlds  at  both  periods  ;  but 
in  measuring  the  average  tallness  of  people  at  diiferent  times 
or  places,  we  are  expressly  dealing  Avith  diiferent  peoples,  and 
so  must  average  their  heights  at  each  period  or  ])lace  se])a- 
rately,  weighting  each  operation  according  to  the  numbers  of 
persons  measured,  and  then  merely  compare  the  results  of  the 
separate  averagings — in   all   cases  using  a   sup])osedly  constant 


200  MA  rilKMATICAL    FOR.MI' LAIK  ».\ 

lucnsnre  of  length  nlrcadx'  cxaniiiicd  and  approNcd.  Again,  to 
use  an  example  nearly  allied  toonrown  subjeot,  in  the  measure- 
ment of  the  average  wealth  of  a  country  at  different  |)erio(ls  we 
must  take  into  account  the  different  total  wealths  and  the  dilicr- 
€nt  numbers  of  the  ]K)pulations  possessing  them.  Here,  for  in- 
.stance,  it  would  he  perfectly  proper  for  the  result  to  indicate 
that,  though  there  be  no  variation  whatsoever  in  the  wealth  of 
the  individuals  in  the  ditferent  classes  of  society  (as  found  by 
<uir  using  a  measure  of  wealth  already  examined  and  a])proved), 
^•('t  if  the  wealthier  classes  increased  more  in  numbers,  the  aver- 
iige  wealth  of  the  (lountry  would  increase,  and  if  the  poorer 
classes  increased  more  in  numbers,  the  average  wealth  of  a 
<'ounti"y  would  decrease. 

Not  so,  however,  in  regard  to  the  exchange- value  of  mcmey 
( \\  Inch  is  the  measure  itself  used  in  the  preceding  measurement). 
Suppose  a  creditor  should  argue  thus  :  "  Prices  have  not  varied 
i\  |)article,  but  I,  and  the  whole  community  as  well,  are  now  ])ur- 
<'hasing  more  expensive  articles  and  fewer  chea])  articles  than  we 
used  to  do  ;  therefore,  as  we  get  a  smaller  (juantity  of  goods  for  the 
sauu'  money,  our  money  has  fallen  in  purchasing  j)owcr  or  ex- 
change-value, and  the  average  of  prices  has  risen,  and  my  debtor 
ought  to  pay  me  back  more  money  than  h(>  borrowed,  to  make 
lip  the  deficiency."  It  is  plain  that  he  would  be  talking  non- 
sense, because  the  debtor  could  readily  reply  :  "  If  you  and  1 
find  the  rest  of  us  s])end  our  money  more  on  ex])ensive  articles 
iuid  less  on  cheap  articles  than  we  used  to,  the  fact  that  prices 
have  not  changed  shows  that,  though  wc  arc  getting  smaller 
•(juautities,  we  are  getting  better  (pialitii's,  and  the  gain  in  the 
latter  makes  up  for  the  loss  in  the  former  ;'"  wherefore  there  is 
no  reason  Avhy  I  should  give  von  back  more  nionev  to  enable 
you  to  get  the  same  (piantit\' of  better  articles." 

Now  if  we  already  had  a  correct  measure  of  exchange-value, 
if  we  wanted  to  measiu'c  variations  of  the  average  precinia^ness, 
not  of  money,  but  ofail  connnodities  in  general,  wc  should  pursue 
exactly  the  course  advised  by  Drobisch  for  measuring  \ariations 
of  tlie  average  exchange-value  of  monew      It  isperfectlv  jtroper 

i«Cf.  Chapt.  rv.,  Sect.  V.,  Xutc  in. 


ERUons   iNcrRiJT-::)  201 

that,  the  ])reciousn ess  of  in(livi(hial  tliin}i;s  rem:iiniii*i  imehanged 
(all  prices  being  constant),  an  increase  in  the  (juantity  of  cheap 
things  should  lower  the  average  of  })recionsness,  and  an  increase 
in  the  quantity  of  costly  things  should  raise  it.  The  three  ab- 
surdities in  Drobisch's  method  as  a  measure  of  the  exchange- 
value  of  money  cease  to  exist  in  it  as  a  measure  of  tiie  j^recious- 
iiess  of  commodities.  We  have  seen,  also,  that  Drobisch's 
method  gives  two  results,  according  as  it  is  a])])liedto  a  common 
weight-unit  or  to  a  common  capacity-unit.  This  is  perfectly 
proper  in  the  method  as  a  method  of  measuring  the  ])reciousness 
of  goods,  because  there  are  two  ways  of  conceiving  of  precious- 
ness.^'  Thus  Drobisch  has  hit  one  thing  while  aiming  at  another. 
But  the  proviso  stated  at  the  commencement  is  of  importance. 
For  the  perfect  working  of  this  method  of  measuring  variations 
in  the  preciousness  of  goods  the  exchange-value  of  money,  by 
which  w^e  measure  the  preciousness  of  individual  things,  ought 
to  be  constant,  and  we  should  know  this.  '  -  Otherwise  its  varia- 
tions must  be  allowed  for  ;  for  which  purpose  also  its  variations 
need  to  be  known.  Thus  the  pr(»])er  method  of  measuring 
variations  in  the  exchange-value  of  money  is  a  prere(piisite  even 
for  the  employment  of  Drobisch's  method  as  a  method  for  meas- 
uring variations  in  the  preciousness  of  all  commodities. 

§  6.  Geometric  averagiuf/  of  prices. — As  in  the  case  of  aver- 
aging the  exchange-values  of  money,  the  geometric  average  of 
prices  does  not  fall  into  two  different  methods  according  as  we 
compare  averages  of  prices  or  average  the  variations  of  prices. 
What  is  an  apparent  weighting  in  the  one  is  actually  the  weight- 
ing both  in  it  and  in  the  other.  If  we  average  i)rices  at  two 
periods  separately  by  using  weighting,  the  same  in  both  cases, 
according  to  the  relative  mass-quantities,  the  same  result  is  ob- 
tained as  if  we  employed  the  same  weighting  in  averaging  the 
variations  of  prices.  But  we  know  that  we  do  not  want  to 
weight  either  the  prices  or  the  variations  according  to  the  mass- 
quantities — or  the  mere  numbers  of  times  some  mass-imits  hap- 
pen to  recur, — but  according  to  the  total  exchange-values,  which 

^'See  above  Sect.  III.,  Note  3. 

^'^  Money  serves  here  as  water  does  for  measuring  the  specific  gravity  of  bodies. 


•202  M.\'riii:MA'ii<  Ai,   foumila  tjon 

correspond   to  the  miinlxTs  ot"  times  e(|uivalent  inass-units  are 
repeated.      Heneo  the  averajie  shoidd  be  in  tliis  form, 

i:;=p'(:;)- (:!:)' (^y — ■ 

in  which  n"  =  a  +  b  +  C  +  to  /*  terms,  and  a  =  xa,  b  =  _?/,9, 

and  so  on.      At  whieli  pei'iods  the  terms  xa,  yj9,  sliould  be 

chosen,  we  have  examined  in  the  |)receding  Chapter,  and  have 
conehided  that  the  best  weighting  is  tor  a  to  be  equal  to  I  x^a^x^a^, 
b  to  1' yi/^jj/o/^jj  swd  so  on.  One  r(>ason  why  weighting  accord- 
ing to  the  mere  mass-quantities  is  not  to  be  employed  is  because 
this  weighting  \vould  be  accidental  according  to  the  sizes  of  the 
mass-units  used.  Or  if,  to  avoid  this  variability  in  the  residts, 
we  used  the  same  mass-unit  throughout,  there  would  still  be  a 
choice  between  two  results,  on  a  unit  of  weight  or  a  imit  of  ca- 
pacity. And  in  either  of  these  cases,  as,  given  the  total  ex- 
change-values of  tw' o  classes  to  be  the  same,  the  same  mass-unit 
W'ould  l)e  repeated  less  frequently  in  a  more  precious  article  than 
in  a  less  precious  one,  the  weighting  would  dej)art  from  the  pro])er 
w^eighting  inversely  according  to  the  preciousness  of  the  articles. 
§  7.  In  a  series  of  successive  periods,  if  we  employ  even 
weighting  or  the  same  uneven  weighting  throughout,  a  compar- 
ison of  subsequent  ))eriods  all  with  n  first  basic  period,  on  the 
geometric  average,  will  yield  results  which,  compared  with  one 
another,  give  the  same  results  as  when  these  periods  are  directly 
compared  with  one  another  witli  tiie  same  weighting.  Thus  the 
conq)aris()n  between  the  second  and  the  first  will  be  as  above 
indicated,  that  between  the  third  and  the  first  like  unto  it  with 
change  of  numbering,  and  that  between  the  results  of  these 
comparisons  as  follows,  ^ 


which  rc(hices  to 


ERRORS    INXTRliKl)  20."5 

which  is  the  formuhi  for  comj)arint)'  the  third  period  with  the 
second  period  directly. 

This  agreement  does  not  exist  in  the  comparisons  made  in 
series  constrncted  on  Carli's  or  on  Yonng's  method,  or  in  a  few 
other  ways,  as  in  one  variety  of  Scrope's  method.  The  employ- 
ment of  these  methods  on  the  iisnal  plan  of  comparing  all  later 
periods  directly  with  an  original  basic  period  involves  an  incon- 
sistency, since  the  indirect  comparisons  between  these  later 
periods  yield  a  different  resnlt  from  what  would  have  been 
yielded  by  the  employment  of  the  same  method  of  averaging  in 
a  direct  measurement  between  these  periods,  or  from  what  the 
indirect  measurement  itself  would  have  been  had  a  different  basic- 
period  been  chosen.'^  This  inconsistency  appears  never  to 
have  been  perceived ""  until  it  was  pointed  out  a  few  years 
ago  by  Professor  Westergaard.  Professor  Westergaard  also  as- 
serted, in  a  general  way,  that  this  inconsistency  does  not  occur 
in  the  geometric  method,  and  advanced  this  difference  as  an  ar- 
gument in  favor  of  the  geometric  method.-'  Credit  is  due  to 
Professor  AVestergaard  for  calling  attention  to  this  difference. 
But  his  use  of  it  as  an  argument  for  the  geometric  averaging  of 
price  variations  suggests  the  following  remarks. 

First,  there  are  other  methods  beside  the  geometric  that  share 

with  it  freedom  from  this  inconsistency.     Thus  in  a  series  in 

which  the  later  periods  are  separately  compared  with  the  first, 

this  inconsistency  is  not  incurred  by  the  use  of  Dutot's  method, 

by  the  use  of  Scrope's  method  with  the  same  mass-quantities  in 

all  cases  (either  those  of  the  first  or  of  some  other  })eriod,  or  a 

general  average  of  all  the  periods  or  of  some  of  them  or  of  any 

periods),^^  or  by  the  use  of  Drobisch's  method  (provided  it  be 

confined   to  the  same  classes).      Hence  the   argument    for   the 

geometric  average  as  exclusively  possessing  a  certain  excellenc'e, 

is  wholly  invalid.      Professor   Westergaard  had  in    mind   only 

^^  See  above  in  i^  3  in  regard  to  Carli's  metliod.  It  is  plain  that  the  same  holds 
good  in  Young's.     Cf.  Appendix  C,  II.  ^  2  and  III.  ^  2. 

20  A  good  example  of  how  it  has  been  ignored  is  furnished  by  De  Foville.  See 
Appendix  C,  IV.  §  2  (2). 

21  B.  110,  pp.  219-220,  followed  by  Edgeworth,  U.  (m,  p.  38(!,  15.  66,  p.  137. 

22  See  the  formulse  for  a  series  of  index-numbers  using  this  metliod,  in  Appen- 
dix C,  lY.il. 


204  MAIIIK.MA'l'K  AI.     I'OKM  1"  I ,  A'lK  ».\ 

C'arli's  and  Young's  nK-thods  ;  and  his  aruunicnt  is  duv  to  this 
restriction.  As  for  the  methods  Mliich  do  incur  this  inconsis- 
tency, these  are  not  only  (Jarli's  and  Vonuii's,  inelndinti'  Pal- 
grave's  variety  of  the  hitter,  but  also  Paasche's  variety  of 
Scrope's,  Lehr's,  Nicholson's  fin  its  one  original  form),  and  an- 
other which  is  still  to  l)c  d(  scrihcd — in  the  last  three  ])r()vided 
thiy  were  to  be  emj)loye(l  in  the  serial  foi'in  with  dependence 
u[)(»n  a  common  base.--^ 

Secondly,  it  should  be  distinctly  noticed  that  even  the  geo- 
metric average  possesses  this  excellence  only  if  used  with  the 
same  weighting  through  the  whole  series.  But  this  system  of 
weighting  evidently  is  slipshod,  and  cannot  pretend  to  give 
j)erfectly  true  results.  The  true  system  of  weighting  is  to  em- 
ploy special  weighting  in  every  single  comparison,  according  to 
the  conditions  existing  at  the  two  periods  compared.  Now  if 
^ve  employed  such  ])ro[)er  weighting  in  comparing  the  second 
w  ith  the  first,  and  again  in  comparmg  the  third  with  the  first, 
the  Aveighting  would  most  likely  be  different  in  these  two 
comparisons.  Then  a  comparison  of  the  results  so  obtained 
would,  as  in  some  of  the  arithmetic  averagings  already  noticed, 
be  a  comparison  between  these  two  periods  with  double  weight- 
ing— and  of  such  a  kind  as  to  lead  to  the  three  absurdities  into 
which  Drobiseh's  method  fell.  And  n(»w  the  direct  comj)arison 
between  the  two  periods,  with  single  weighting,  would  be  with 
weighting  still  different  from  that  used  in  either  of  the  preced- 
ing comparisons,  and  its  result  would  not  agree,  except  acciden- 

-■■'In  all  these  methods  the  direet  comparisons  between  the  later  periods,  and 
the  indirect  comparisons  l)et\veen  them  through  mediation  of  their  comparisons 
with  the  basic  period,  may  agree  under  certain  conditions,  it  is  easy  to  discover 
these  conditions  by  comparing  the  formulaj,  and  it  maybe  interesting  to  state 
them.  Tims  the  indirect  agrees  witii  tiie  direct  comparison  in  Carli's  method  if 
between  the  basic  and  tiie  nearest  subsequent  period  (or  furtiiest  prior  period) 
tliere  be  no  price  vftriations,  or  if  all  these  be  in  the  same  proportion  ;  in  Voidkj's 
metiiod,  the  same  ;  in  Pdlgrarc's  variety  of  Young's  method,  if  the  mass-quantities 
be  the  same  at  the  two  sul>se(iuent  periods,  or  if  they  have  varied  between  these 
two  periods  in  the  same  pr(>])(>rtion  ;  in  J'aasc/te's  variety  of  Scrope's  method,  if 
between  the  two  sul)se(|ucnt  periods  tliere  be  no  variations  either  in  the  prices  or 
in  the  mass-(iuantities,  or  if  between  these  two  periods  both  the  prices  and  the 
iiiass-(iuantiti(!S  have  all  varied  in  one  and  the  same  projiortion.  The  other 
iiu'tiiods  iiave  never  Iteen  suggested  for  use  in  this  serial  form.  What  would  be 
tiie  conditions  of  such  agreement  in  them  if  so  used  will  lie  examined  for  another 
purpose  in  a  later  ('iiaptcr. 


EER(J1!8    IN<  I  i;i!i:i)  20") 

tally,  witli  the  result  obtained  hy  eoniparin^i'  tlieir  results.  Yet, 
supposing  the  geonietrie  average  the  i-ight  one  to  use,  among 
these  comparisons  it  is  only  this  <lireet  one,  if  the  ])erio(ls  be 
contiguous,  that  can  give  the  correct  result. 

Thirdly,  it  would  seem  to  be  unquestionalile  that  the  agree- 
ment between  the  indirect  and  the  direct  comparisons  ought  to 
exist.  Professor  Westergaard  based  his  recommendation  of  the 
geometric  average  u})on  the  princi])le  that  prices  measured  from 
1860  to  1870  and  from  1870  to  1880  ought  to  show  the  same 
variation  from  1860  to  1880  as  would  be  shown  by  comparing 
the  prices  of  1880  directly  with  tliose  of  1860.^*  By  adding 
another  supposition  this  can  be  made  self-evident.  Su])])ose 
thai  everything  at  the  last  period  is  in  exactly  the  same  stiite  as 
at  the  first  (or  that  everything  diverges  in  the  same  proportion). 
Then  it  is  evident  that  the  exchange-value  of  money  is  the  same 
at  the  last  jx'riod  as  at  the  first  ]>eriod  (or  deviates  inversely  to 
a  uniform  price  variation),  whatever  intervening  variations  may 
have  taken  place.^^  Constancy  (or  that  variation)  is  shown  by 
the  direct  comparison  between  these  two  periods  ;  and  constancy 
(or  that  variation)  should  be  shown  by  a  series  of  comj)arisons 
through  intervening  periods,  no  matter  how  many  or  how  varied 
the  intervening  periods  may  be.  But  it  does  not  follow  tliat  this 
agreement  should  be  sought  at  the  ex})ense  of  other  rcipiire- 
ments,  or  that  such  agreement,  if  existing  in  any  method,  can 
prove  it  to  be  true,  if  we  already  know  that  this  methcjd  fails  in 
other  respects.  For  instance,  Drobisch's  method  ])rovides  this 
agreement,  but  Drobisch's  method  is  not  thereby  proved  to  be 
correct.  The  need  of  this  agreement  can  be  taken  only  as  a 
negative  test,  or  touchstone.  If  a  method  has  other  things  to 
recommend  it,  it  may  be  disproved  by  being  found  not  to  stand 
this  test.  But  unless  it  is  recommended  by  other  reasons,  it  is 
not  proved  by  its  satisfying  this  test.      Perhaps,  however,  the 

24  B.  IKf,  p.  219. 

25  This  would  seem  to  be  so  also  if  only  all  the  prices  were  the  same  (or  equally 
divergent)  at  the  separate  periods,  let  the  mass-quantities  be  what  they  niay^n 
accordance  with  Propositions  XXVII.  and  XVII.,  and  XLIV.  and  XLV.  But 
if  the  mass-quantities  as  well  as  the  prices  are  all  exactly  the  same  (or  equally 
divergent),  there  can  be  no  question  about  it. 


20()  MA  Til  i;m.\  ricAi.   KoitMn, a  tion 

peii'c'ctly  correct  method,  lor  a  loiii:'  sccjuciicc  ot"  periods,  will  not 
be  found  to  Ue  workable,  and  the  workable  form  that  comes  the 
nearest  to  it  may  not  satisfy  this  requirement  perfectly.  Tlien 
this  requirement  may  be  used  as  a  means  of  correetino;  the  latter. 

Lastly,  this  arji:ument  of  Professor  AVesterj^aard's  leads  to  the 
consideration  whether  we  should  continue  in  the  usual  coui'se  of 
comparing-  every  suhsecpient  ])eriod  with  the  same  oriirinal 
period  as  common  basis  for  a  whole  series,  or  whether  the  series 
should  be  formed  by  comparing  every  period  with  the  immedi- 
ately precedinu'  |»erio(l,  and  again  the  next  period  witli  it,  and 
welding  tliem  all  togethei'.  The  latter  is  as  praetical)le  a  course 
as  the  former,  and  may  yield  the  same  sort  of  serial  index-num- 
bers. For  exanq)le,  if  we  find  tlie  result  of  the  first  comparison 
to  show  that  between  the  first  and  the  second  periods  prices  have 
fallen  on  the  average  by  10  per  cent.,  and  the  result  of  the 
second  comparison  to  sho>\'  that  between  the  second  and  the 
third  periods  they  have  risen  by  oO  ])er  cent.,  we  may  express 
tiiese  relations  in  the  series  100,  90,  120,  which  indicates  at 
once  that,  by  passing  tlirough  the  second  jx'riod,  the  level  of 
prices  at  the  third  is  2(»  per  cent,  above  what  it  was  at  the  first. 
As  to  tiie  first  of  these  procedures,  there  is  little  in  its  favor  but 
its  convenience.  The  most  natural  and  rational  ))rocedure,  the 
moment  we  attach  importance  to  correct  weighting,  would  seem 
to  !)('  the  second.  At  first  sight  the  argument  between  tlie  two 
ai)pears  to  lie  merely  between  the  opinions  that  in  the  former  we 
gain  exactness  in  the  comparisons  of  the  later  periods  with  the 
first  and  lose  it  in  the  comparisons  between  the  later  periods, 
and  in  the  second  we  gain  exactness  in  the  comparisons  between 
contiguous  |)ei'iods  and  lose  it  in  all  other  compariscms.  But  of 
(M»ui'se  both  these  positions  are  false,  being  self-contradictory. 
The(|uestiou  is  really,  In  which  of  the  two  courses  are  we  likely 
to  gain  greater  exactness  in  the  (ronq)arisons  actually  made? 
Here  the  probability  seems  to  incline  in  favor  of  the  second 
course  ;  for  the  conditions  are  likely  to  be  less  diverse  between 
two  contiguous  periods  than  between  two  periods  say  fifty  years 
apart. 

5j  <S.   There   is  still   another   reason    for  adopting   the   second 


EUHOKS    INcrRRHl)  207 

course.  This  is  that  the  first  course  is  tied  to  the  employment 
of  a  definite  number  of  chisses,  decided  once  for  all  at  the  start ; 
but  the  second  course  permits  the  dropping  at  any  time  of  an 
old  class  or  the  introduction  of  a  new  one.  In  fact  this  second 
course  is  the  only  one  permittini;:  an  accurate  measurement  of  the 
exchange- value  of  money  over  a  very  long  series — say  of  one  or 
several  centuries.  For  in  ])eri()ds  separated  by  a  very  long  in- 
terval there  is  much  that  is  not  common  to  both  the  periods,  and 
to  comjjare  them  by  what  is  connnon  to  them  both  is  to  make  a 
restricted  and  ineomjilete  comparison.  A  complete  comparison 
can  be  made  only  by  comparing  all  that  is  in  each  of  them  with 
the  intervening  ])eriods  near  to  them,  gradually  dropping  out  the 
old  things  as  they  disappear  and  introducing  the  new  as  they 
appear.  It  is  also  only  in  the  comparison  of  such  temporally 
distant  periods  that  the  divergence  between  the  two  ways  of 
conducting  the  measurement  is  likely  to  be  great.  Therefore 
for  k>ng  series  it  is  important  to  decide  between  the  two  courses. 
Now  if  the  two  periods  be  two  years  separated  by  a  century,  it 
is  probable  that  if  a  rightly  conducted  measurement  had  been 
made  of  the  variations  from  year  to  year  filling  up  the  whole 
century,  the  result  at  the  end  would  be  very  nearly  the  true  one, 
while  a  direct  com])arison  between  the  two  extreme  years,  which 
comparison  might  have  to  leave  out  altogether  some  classes  that 
have  been  counted  in  the  other  comparisons,  because  they  did 
not  exist  at  the  first  period  or  no  longer  exist  at  the  last,  might 
be  appreciably  divergent  from  the  truth,  even  though  the  com- 
parison were  made  in  jiroper  manner  and  the  data  were  correct. ^^ 
Hence  the  procedure  of  comparing  every  successive  period  with 
the  immediately  preceding  is  the  proper  one  to  adopt  if  it  be 
intended  to  continue  the  measurement  over  many  years,  or  in- 
definitely."" 

Tncidentally  we  hereby  see  alsd  that  among  the  reasons  why 
we  cannot  successfully  com])are   the  exchange-value   of  money 

26  Cf.  above  Chapt.  IV.,  Sect.  V.  j;  7,  Note  IS. 

2'  Into  this  course  De  Foville  was  forced  by  circumstances  in  his  calculations 
after  1862.  It  has  been  reconiniended  by  Lehr  for  his  method,  B.  68,  pp.  45,  46, 
and  ill  general  by  the  British  Association  Comniittec,  First  Report,  B.  99,  p.  250, 
and  PMgeworth,  B.  .59,  pp.  268-269. 


208  MAIIIKMATICAL    F(  )|;M  T  I.AIK  )N 

to-Jay  with  what  it  was  Hvc  hundred  years  a»;()  is  not  nierelv 
that  we  have  not  aeeurate  (hita  altout  the  year  14()(),  l)ut  that  we 
have  not  complete  data  about  all  the  intervening  years.  Yet 
there  is  no  reason  why,  if  from  now  on  the  measurement  be  cor- 
rectly made  from  year  to  year,  the  people  living  in  2400  should 
not  know  with  almost  perfect  exactness  the  relation  between  the 
exchange-value  of  their  money  and  ours,""* 

The  l)rief  hy])othetical  series  of  index-numbers  above  used  as 
examples,  namely  100,  JH),  120,  can  be  reduced  to  this  :  111.11, 
100,  133.33,  or  again  to  this  :  83.33,  75,  100,  or  to  any  other 
.series  in  wliich  the  figures  observe  the  same  proportions.^  This 
fact  shows  that  it  is  of  no  importance  what  period  is  selected  as 
the  so-called  basic  period  of  the  series,  wdiichever  of  the  above 
two  courses  be  adopted.  Properly  every  period  is  basic  in  the 
comparison  with  the  next  period,  and  none  is  basic  for  a  series 
of  periods.  A  series  with  the  index-number  100  at  one  period 
does  not  better  represent  the  sequence  of  events  than  a  series 
with  this  index-number  at  any  other  period.  Xor  is  there 
rhyme  or  reason  for  making  the  basic  period  longer  than  the 
other  periods.  Much  discussion  as  to  the  superiority  of  this  or 
that  base  in  the  various  attempts  at  measuring  the  course  of  the 
exchange-value  of  money  during  the  latter  half  of  the  century 
just  elapsed  has  been  wasted.*' 

YTT. 

§  1.  There  remains  only  to  point  out  the  mathematical  formu- 
hition  foi'  the  exchange-value  of  inon(y  in  a//  things  {i)V  inversely 
for  the  level  of  all  prices,  including  the  price  of  money).     This 

-^  The  not  infreiiuent  complaint  about  the  ineomiuensurability  of  the  oxohange- 
valiie  of  money  at  two  long  separated  periods  directly  compared  {e.  g.,  recently 
by  W.  Cunningham,  On  the  value  of  money,  (Quarterly  .Journal  of  Economics, 
.July  1899,  pp.  ;i79-385)  should  be  allowed  no  influence  in  the  question  as  to 
whether  the  variation  of  this  exchange-value  tlirough  successive  periods  is  meas- 
urable. The  dirticulty  is  greater  in  the  attemjjt  to  measure  the  contemporaneous 
levels  of  the  exchange- value  of  money  in  two  distinct  countries,  say  an  arctic  and 
a  tropical.     Hut  luckily  we  have  not  so  much  interest  in  measuring  this. 

2'-*  The  table  of  lOvelyn  on  the  basis  of  lo5()  was  thus  turned  about  and  put  on 
the  basis  of  1700  by  .J.  P.  Smith,  B.  7,  pp.  472-470. 

^^  Concern  for  the  base  was  first  shown  by  Porter,  15.  11,  p.  440.  That  the  base 
should  be  carefully  selected  istiie  first  of  thecaiums  laid  down  by  Martin,  op.  cit., 
]).  f)2(i.     It  is  said  to  be  of  great  importance  l>y  Mayo-Smitii,  B.  1.57,  p.  48(). 


OF    KX('nAN(iK-VALri-;    IN     ALL     rillNCS  20!) 

is  easy  on  the  assumption  that  any  of  the  mathematical  averages 
is  })roper  for  ex]>ressing  the  exchange-vahie  of  money  in  all  other 
things.  We  simply  liave  to  add  to  the  riglit  side  of  the  ecpia- 
tion  the  expression  for  tlie  exchange-value  of  the  money-unit  in 
money,  -vvhicli  is  always  unity,  and  invariable. 

Thus,  taking  the  arithmetic  average  and  using  M^,  to  express 
the  exchange-value  of  the  money-unit  in  all  things,  we  have  for 
the  first  period,  with  even  weighting, 

Ma^=---^{1  -H  {aA)  +  (hB)  +  (cC)  +  to  n  -\-  1  terms}, 

which  reduces  to  unity.  And  for  the  second  period,  again  Avith 
the  same  even  weighting,  we  have 

^^«^  =  .T^^ri  i  ^  +  «'(«^^)  +  ^'(^^)  +  ''('^) 

-)-  to  n  -|-  1  terms}. 

And  the  comparison  of  these  two  reduces  to 

M  1 

\/-  =  -  r-M  +  a'  +b'  +  c'  +  to  n  +  1  terms).      (18,  i) 

For  the  other  kinds  of  averaging,  still  with  even  weighting, 
we  should  have,  after  similar  reductions, 

^2_ 1 .,.       . 

Mar~       1       r         111  ,  V   ^        ^^ 

— r- 1  I  1  +  — ,  +  n  +  -/  + to  n  -I-  1  terms  ) 

n  +  i  \         a        b'      c'  J 

and 

Ma^  _  «+;/«'  -b'   c'  ■ to  n  terms.  (18,  A 

M         ^ 

It  is  easy  to  restate  these  formuhe  with  the  terms  — ,  7^,  — , 

to  represent  the  variations,  in  place  of  «',  b' ,  c' , 

Inversely,  by  means  of  averages  of  prices,  the  same  results 
may  be  obtained  by  inserting  in  the  formulae  for  the  averages  of 
price  variations  the  price  also  of  the  money-unit,  which  is  in- 
variably a  unit. 

With  uneven  single  weighting,  we  have  to  add  the  weights  of 
14 


210  MATllKMATK  AI,     FORM  I '  LA  IK  »N 

the  c'(tiniii(»(lity  classes  a.s  bcture,  and  in  addition  we  must  give 
some  Aveiglit  to  the  class  money.  AVhat  the  nature  of  this  weight 
is,  will  be  left  over  for  a  later  discussion  (in  Chapter  XIII.). 
Here  we  may  rej^resent  it  by  m.  Then,  using  the  form  for  the 
variations  of  [)riees,  with  P„  to  represent  the  level  of  prices  with 
the  ])rice  of  )non(y  included,  we  have 

^^  —— L ,  (19,  ,) 

^"'         „\        fm  +  a^  +  b  ^+C^-i+ to«  +  ltermsV 

n"  +  m  V  "2         ;i,         :  2  / 

J''=  -„  (m  +  a  — +  b9^  +C^  + to  ?i  +  1  terms  )  ,    (19,2) 


^rvi^-m-^ ™ 

It  is  evident  on  inspection  that,  compared  with  the  results  of 
the  formnlse  for  the  corresponding  averages  of  the  variations  of 
money  in  exchange- value  in  all  other  things,  the  results  of  these 
formnla?  always  show  a  smaller  variation  than  those,  in  agree- 
ment with  Proposition  XXI.  ;  that  they  always  indicate  con- 
stancy when  those  do,  in  agreement  M'ith  Propositions  XXII. 
and  XXVI.  ;  that  they  always  vary  in  the  same  direction,  in 
agreement  w  ith  Proposition  XXIII.  ;  that  their  divergence,  or 
falling  short,  from  those  is  ahvays  smaller  the  gi"eater  the  num- 
ber of  terms  (or  rather,  the  larger  is  n"  compared  with  m),  in 
agreement  with  Pro])osition  XXl  \'.  ;  and  tliat  the  |)ro|)ortion 
of  this  divergence  is  always  the  same,  in  agreement  with  Propo- 
sition XXV. ;  and  lastly  that  their  indications  are  unaffected  by 
the  numbers  of  other  things  that  are  constant  in  general  ex- 
change-value, if  the  indications  l)e  of  constancy,  or  that  vaiy 
in    tile   same   j)roportion    in    exchange-value  in  all  things  (their 

prices  varying  as  ,j""  J,  in  agreement  with  Propositions XX XII., 

XXXV.  and  XXXVI. 

§  2.  Other  formula?  may  be  altered  in  the  same  manner.  In 
Scrope's  method  the  want  of  body  to  money — the  indifference 
we  all  feel  to  its  mass — causes  no  trouble,  because  all  the  terms 
are  the  total  money-values  of  the  classes,  and  so  long  as  we  can 
get  this  in  the  case  of  money,  without  reference  to  its  weight  or 


OK    EX('HAN(ii;-\ALrE    IN    ALL    Tin.\(;.S  211 

bulk/  we  aiv  satisfied.  Tims,  witii  r  representing  the  number 
ol'  money-units  employed,  the  formula'  for  Scrope's  method  are 
the  following, 

and 

^^"'  _  ''  +  ''''^1  +  ■''''^  +  •-'^'■+ 

The  same  remarks  apj^ly  to  these  as  have  just  been  made  on  the 
formulae  for  averaging  the  ])rire  variations  (except  the  reference 
to  Proposition  XXV.). 

In  Drobisch's  method,  however,  where  specific  weights,  or 
capacities,  appear  alone  in  one  half  of  the  formula,  it  would  be 
impossible  to  make  the  alteration  permittmg  this  method  to  be 
applied  to  tlie  measurement  of  the  exchange-value  of  money  in 
all  things.  Here  is  one  more  reason  showing  the  falsity  of  that 
method. - 

But  there  are  methods  ('m])loying  double  weighting  that  are 
applicable  to  the  measurement  of  the  exchange-value  of  money 
in  all  things,  including  itself. 

''■  The  mass-unit  of  luoney  which  we  habitually  use  is  nothing  else  than  the 
money-unit  itself,  whose  money-value  is  always  one  money-unit. 

-  As  a  measure  of  the  preeiousness  of  commodities  it  is  no  fault  in  this  method 
that  it  cannot  be  applied  to  all  things,  including  money.  For  money,  in  which 
mass  is  inessential,  has  no  preeiousness. 


CHAPTER  YT. 

THE  QUESTION  OF  THE  MEANS  AND  AVERAGES. 

I. 

§  1 .  We  are  now  ])repare(l  for  entering-  upon  our  .subjt'ct  ])roper, 
the  search  after  the  right  method  of  measuring  general  exchange- 
value  and  its  variations.  We  must  deal  first  exhaustively  witli 
exchange-value  in  all  other  things,  as  the  simpler. 

When  one  thing  rises  or  falls  to  a  certain  extent  in  exchange- 
value  in  every  other  thing  alike,  we  know  (bv  Proposition  XVII.) 
that  it  rises  or  falls  to  the  same  extent  in  exchange-value  in  all 
other  things,  and  (by  l*roposition  XVIII.)  that  every  one  of  the 
other  things  have  reversely  fallen  or  risen  to  a  smaller  extent  in 
exchange-value  in  all  other  things.  We  also  know  (by  Prop- 
osition XIX.)  that  the  larger  the  number  of  the  things,  the 
smaller  is  the  opposite  variation  of  every  one  of  the  other  things. 
But  we  do  not  "as  yet  know  how  great  the  op])osite  variation  of 
every  one  of  the  other  things  is,  given  the  variation  of  one  and 
their  number.  Here  is  a  Proposition  lacking,  whicli  if  we  siiall 
be  able  to  su])ply,  we  shall  be  able  to  measure  all  exchange-value 
variations.      The  ])robl('m,  then,  is  to  supply  this  Proposition. 

Jn  attempting  to  solve  a  problem  it  is  well  to  put  it  in  its 
simph'st  form.  Tlie  simplest  form  in  which  we  can  put  our 
problem  is  the  following.  Let  us  su])posc  tliat  we  arc  dealing, 
with  a  world  in  which  are  only  three  classes  of  exchangeable 
things  (or  with  a  |)art  of  our  world  which  consists  of  only  three 
classes).  One  of  them  let  us  suppose  to  be  money,  so  that  we 
have  a  woi'ld  with  money,  represented  by  [^I],  and  two  classes 
of  commodities,  rej)resented  by  [A]  and  [B] .  IIci'c  all  the 
other  things  beside  tlie  one  class,  money,  whose  exchange- value 
we  are  examining,  are  comprised  in  only  two  classes.     These 

212 


TlIK    I ' KOI ; I. KM    AM)    ,Sl'(JGE«TEI)    SOLUTIONS  21. "3 

two  classes  of  commodities  we  may  also,  for  simplicity,  suj)pose 
to  be  equally  important,  or  equally  large — not  at  one  period  only, 
but  either  at  each  of  the  periods  compared  or  somehow  over  both 
the  periods  together,  without  reopening  here  the  question  how 
their  importance  or  economic  size  is  to  be  computed.  Now  if 
we  succeed  in  solving  the  })roblem  in  this  confined  and  simple 
form,  we  shall  not  be  able  immediately  to  extend  the  same  solu- 
tion to  all  cases ;  for  Avhat  is  true  of  two  equally  important 
classes  may  perhaps  not  admit  of  being  applied  to  more  com- 
plex cases.  But  if  we  succeed  in  the  simplest  case,  we  shall  be 
well  on  the  way  to  solving  the  complex  cases.  These,  more- 
over, may  be  admitted  to  consideration  as  w'e  advance  further, 
even  before  we  finally  solve  the  simple  cases. 

Simplicity  in  the  form  of  the  problem  will  also  be  carried  out 
by  supposing  that  at  the  first  period  the  money-unit  is  equiva- 
lent to  a  mass-imit  of  each  of  the  commodities,  so  that  M  c=A-o=B, 
and  the  prices  of  A  and  B  are  one  money-unit  each.^  We  may 
then  suppose  that  at  the  second  period  A  rises  in  exchange- value 
in  the  other  two,  individually  and  together,  by  one  half,  or  by 
50  per  cent.  ;  which  means  that  its  price  rises  to  1.50,  w^hile  the 
price  of  B  remains  1.00.  At  the  same  time  M  and  B  have 
each  fallen  by  one  third  in  exchange-value  in  [A],  or  by  33 J 
])er  cent.,  but  as  ^I  retains  its  exchange-value  in  [B]  and  B  its 
in  [M],  ^I  has  not  fallen  as  much  in  [A]  and  [B]  together, 
that  is,  in  exchange-value  in  all  other  things,  nor  has  B  fallen 
so  much  in  [A]  and  [M]  together,  that  is,  also,  in  exchange- 
value  in  all  other  things. 

Digressively  it  may  here  he  remarked  that,  as  a  corollary  to 
Proposition  XIY.  [M]  and  [1>],  that  is,  two  or  more  classes 
that  retain  the  same  exchange-value  in  each  other,  vary  alike  in 
exchange-value  in  all  other  things  beside  themselves  and  in  exchange- 
value  in  all  things  including  themselves,  but  not  in  all  other  things 
(?.  e.,  other  to  each  class),  unless  these  classes  be  equcdly  large 
{over  both  the  periods   together),    (Proposition   XLVI.).      The 

^  But  -Nve  must  be  on  our  guard  not  to  regard  A  and  B,  equivalent  at  the  first 
period  only,  as  the  economic  individuals  in  the  classes  [A]  and  [B]  for  the  two 
periods  compared  ;  for  the  economic  individuals  must  be  equivalent  over  both 
the  periods  together. 


214         THE    (iUESTIOX    OF    TIIK    MEANS    AND    A\K1{A<;ES 

reason  why  tlu-y  do  not  necessarily  vary  (or  remain  constant) 
alike  in  exchange-value  in  all  other  things,  is  becanse  for  each 
the  standard  of  other  things,  composed  of  the  classes  other  to  it, 
is  different  from  Avhat  it  is  for  every  other.-  Thus,  in  the  ex- 
ample before  us,  if  [M]  be  a  class  twice  as  large  as  [B],  and 
consequently  of  [A] ,  the  rise  of  A  will  have  less  mfluence  in 
depressing  the  exchange-value  in  all  other  things  of  [B]  than  in 
depressing  the  exchange- value  in  all  other  things  of  [A],  since 
it  is  a  rise  of  only  one  third  of  the  things  other  to  the  class  [B], 
while  it  is  a  rise  of  one  half  of  the  things  other  to  the  class  [M] . 
But  if  the  classes  [M]  and  [B]  are  equally  large  or  important, 
the  class  [A]  will  be  of  the  same  relative  size  among  the  things 
other  to  each  of  them,  and  hence  will  have  the  same  influence  in 
making  them  vary  in  exchange-value  in  all  other  things.  We 
do  not  here  concern  ourselves  with  the  size  of  the  class  [M], 
and  confine  our  attention  to  enquiring  what  influence  upon  its 
exchange-value  in  all  other  things  has  a  rise  of  the  exchange- 
value  of  [A]  both  in  it  and  in  [B]  .  For  this  purpose  we  must 
presuppose  knowledge  of  the  relative  sizes  of  [A]  and  [B]  ; 
and,  to  repeat,  the  simplest  case  is  to  sn])pose  that  they  are  equally 
large. 

Our  special  problem  then  is  :  Given  the  abo\e  supposed  con- 
ditions and  variations  of  A,  how  much  has  M  fallen  in  exchange- 
value  in  all  other  things,  that  is,  in  [A]  and  [B]  together? 
Now  the  fall  of  ISI  in  [A]  and  [B]  together  is  the  same  as  its 
fall  in  [A]  and  [B]  under  the  su])position  that  A  and  B  both 
rose  together  to  some  extent.  Therefore  this  prol)lem  is  the 
same  as  to  ask  :  To  what  rise  of  A  and  B  toyetJwr  -is  the  rise  of 
A  alone  equal  in  its  influence  upon  the  exclmnge-value  of  JI  in  all 
(both  these)  other  classes  of  things  f  What  we  want  is  to  reduce 
irregular  variations  in  the  particular  exchange-values  of  the 
money-unit  to  a  uniform  variation  of  all  of  them,  because  when 
we  have  the  latter,  we  know  the  variation  in  the  exchange-value 
of  tlie  money-unit  in  all  other  things  (by  Proposition  XVII.). 

§  2.  To  this  problem  an  answer  might  be  made  wliich  lias 
the  nature  of  an  objection.      It  might  be  argued  that,  though  M 

~  Cf.  also  Proposition  XXXIIJ. 


iHF,  ri!()i!i,i:M   and  sicaiKsri-n^  soi.r'iroxs  215 

ha.-^  fallen  from  the  power  of  pureha8in<;  A  to  the  j)o\ver  of  pur- 
chasing only  f  A,  yet  as  this  f  A  has  risen  by  one  half  in  ex- 
change-value in  all  other  things,  therefore  M  still  commands  as 
nluch  exchange-value  in  all  other  things  as  before,  and  still  })os- 
sesses  as  much  ;  and  \vith  ]M  also  B.  But  this  argument  as- 
sumes that  A  has  risen  by  one  half  in  exehange-Nalue  in  all 
other  things  themselves  remaining  constant  in  exchange-value 
in  all  other  things.  This  implied  assumption  is  false.  With 
the  same  evidence  with  Avhich  we  perceive  that  A  has  risen  by 
one  half  in  exchange-value  in  all  other  things,  we  perceive  that 
the  other  things  have  fallen  somewhat  in  exchange- value  in  all 
other  things.  We  know  that  M  has  fallen  as  well  as  we  know 
that  A  has  risen.  And  the  two  statements  are  perfectly  con- 
sistent. For  when  M  purchases  two  thirds  of  A,  which  has 
risen  by  one  half  in  exchange-value  in  all  other  things,  them- 
selves fallen  in  all  other  things,  M  does  not  command  so  much 
exchange-value  in  all  other  things  as  it  did  before. 

Then  another  solution  might  at  once  be  offered,  which,  if  cor- 
rect, would  settle  our  problem  without  further  trouble.  It 
might  be  argued  that  if  at  the  second  period  M  had  become 
equivalent  to  ^(§A  +  fB),  it  would  have  fallen  by  one  third  in 
exchange- value  in  all  other  things  ;  therefore,  since  it  has  fallen 
to  equivalence  with  i(|A  -|-  B),  it  has  fallen  by  one  sixth,  a 
fall  of  one  third  in  one  thing  being  equal  to  a  fall  of  one  sixth 
in  two  things.  But  this  argument  makes  a  mistake  similar  to 
that  in  the  preceding.  That  argument  neglected  to  take  account 
of  the  fall  of  M  and  of  B  in  exchange- value  in  all  other  things. 
This  argument  neglects  to  take  account  of  the  fall  of  B  in  ex- 
change-value in  all  other  things.  M  purchases  only  one  Avhole 
of  an  article  which  has  fallen  with  it.  It  purchases  two  thirds 
of  an  article  which  has  risen  by  one  half  in  exchange- value  in 
articles  which  have  fallen.  Therefore  M  would  seem  to  liave 
fallen  by  more  than  one  sixth.  But  from  these  data  alone  uo 
definite  conclusion  can  be  drawn  concerning  the  extent  of  the 
fall  of  M  (and  of  B)  in  exchange- value  in  all  other  things. 

§  3.  If  we  could  know  how  much  A  has  risen  in  exchange- 
value  in   all   things,  this   would   give  us   sufficient   informtition 


21  <)        TIIK    (il-KSTIOX    OF    THE    MKANS    AND    AVI:1!A<;KS 

from  which  to  calciihite  the  exchange-values  of  ]\I  and  of  !'>. 
But  we  cannot  know  that  until  we  know  how  much  the  ex- 
change-values of  M  and  R  liave  fallen.  Thus  the  heiiiniiino: 
seems  to  be  lacking. 

The  trouble  is,  we  are  as  yet  in  possession  of  only  one  e<iua- 
tion  with  two  unknown  quantities.  We  know  the  equality  of 
the  exchange-value  of  ^I  at  the  second  period  to  the  exchange- 
value  of  §  A  and  of  B.  But  we  do  not  know  the  exchange- 
value  of  B  any  more  than  we  know  tluit  of  M  itself;  and  al- 
though we  know  the  exchange-value  of  A  in  all  other  things, 
that  is,  other  to  it,  this  is  in  another  standard,  and  we  do  not 
know  the  exchange-value  of  A  in  the  standard  of  otlier  things 
for  ]M,  namely  [A]  itself  and  [B],  We  cannot  know  this 
until  we  know  tlie  variation  of  B,  or  of  M  itself,  in  this  stand- 
ard. A  further  piece  of  information  is  wanting.  In  order  to 
solve  the  problem  we  must  sui)ply  ourselves  somehow  w  ith  tliis 
otlu'r  information.  Further  consideration  of  the  (juestion,  with 
variations  in  the  manner  of  putting  it,  may  i)erliaps  yiehl  us 
further  insight  into  the  nature  of  our  subject,  and  so  disclose 
that  we  are  in  possession  of  the  information  desired. 

§  4.  Now  we  do  })ossess  a  piece  of  information,  already 
noticed,  whicli  is  of  some  service  here.  This  is  the  knowledge 
that  the  variation  of  M  (and  of  B)  in  exchange-vahie  in  all 
other  things,  when  A  rises  to  H,  is  less  than  the  inverse  variti- 
tion  from  1  to  -|,  that  is,  it  is  to  some  quantity  between  1  and 
f.^  This  provides  us  with  a  hint.  Between  two  numbers  we 
are  accpiaintcd  with  several  mathematical  means,  and  the  idea 
suggests  itself  that  the  variation  in  (piestion  may  perhaps  be  to 
one  of  the  mathematical  means  between  1  and  H.  This  sugges- 
tion becomes  moi'c  plansil)lc  when  we  recall  that  the  ibrmulse  we 
have  discovered  for  exchange-value  variations  are  expressions  of 
the  three  "  classic  "  means  or  averages.  Thus  if  it  be  true  that 
the  variation  of  3Ifj.^  compared  with  31,,^  is  to  (me  of  these  means 

^According  to  I'ropo.sitioii  WIIJ.  llvidently  to  ])roduce  the  same  influeiiee 
upon  the  exehange- value  of  M  a  eomiiioii  variation  of  A  and  B  together  cannot  be 
80  large  as  tlie  variation  of  A  alone ;  for  if  we  had  the  variation  of  A  alone  in  the 
first  place,  intiueneing  M  to  a  certain  extent,  an  added  variation  of  |{  in  the  same 
direction  would  increase  tlie  influence  ui>nn  M.  ( 'onipaiv  witli  tliis  the  reasoning 
leading  up  to  Proposition  XXX. 


THE    I'lJOHLK.M    AM)    Sl'(r(;i:sTEl>    SOU    TloNS  217 


14  ,       ,  . 

the  arithmetic,  or  to  —t; r^  =  ^(=  .80),  the  harmonic,  or  to 


1  /•-         \       '"> 
hetween  1  and  '^,  it  would  be  either  to  ;^l  :7  +  '  I  =  t.(=  •^^'^>''^>), 

/2^ 
1/    ^  ■  1  =  .8165,  tlie  geometric  ;  whence  the  percentage  of  tlie 
o 

fall  may  be  calculated,  in  hundredths,  by  subtracting  the  result 

from  unity.     To  be  sure,  we  have  here  only  a  hint,  but  it  is 

sufficient  to  incline  us  to  try  these  suggested  answers,  and  so  at 

once  to  narrow  our  eiKjuiries  ;  for,  although  there  are  other  more 

complex  mathematical  means,  we  need  not  notice  them  unless 

the  true  answer  be  not  found  among  the  simpler  ones.^ 

And  this  hint  extends  to  the  average  of  prices  also.  The  re- 
lationship between  these  and  the  averages  of  the  exchange- 
values  of  M  Ave  have  already  seen.  Thus  in  our  suppositional 
case  the  recij)rocal  of  the  arithmetic  mean  between  1  and  f , 
namely,  .8333,  is  1.20,  which  is  the  harmonic  mean  between 
1.00  and  1.50  ;  the  reciprocal  of  the  harmonic  mean  between 
those  exchange-values,  namely  .80,  is  1.25,  which  is  the  arith- 
metic mean  between  these  prices  ;  and  the  reciprocal  of  the  geo- 
metric mean  between  those  exchange-values,  namely  .8165,  is 
1.2247,  which  is  likewise  the  geometric  mean  between  these 
l)rices.  Therefore  if  we  find  to  which  average  between  1  and  f 
the  exchange-value  of  M  in  all  other  things  has  sunk  Avhen  A 
alone  rises  to  1|,  we  shall  know^  to  which  average  between  1.00 
and  1.50  the  rise  of  both  A  and  B  together  is,  in  its  influence 
upon  ^I,  equal  to  the  rise  of  A  alone  to  1.50. 

Thus  our  simple  problem  becomes  a  question  of  mathematical 

means.     And  as  it  has  been  stated  in  the  simplest  way  possible, 

it  has  l)een  made  to  employ  single  weighting.     The  question  has 

been  })ut  as  a  question  of  means  between  variations  of  exchange- 

'  The  reader  cannot  be  too  often  warned  that  we  are  not  averging  i^A  and  IB, 
l)ut  -;.l  and  1J5 — we  are  not  averaging,  for  instance,  ?,  pound  of  sugar  and  1  pound 
of  copper,  which  would  be  meaningless,  but  we  are  averaging  the  exchange-value 
of  M  in  [A],  say  sugar,  and  the  exchange- value  of  M  in  [B],  say  copper,  in  eadi 
case  measured  at  the  second  period  by  comparison  with  what  it  was  at  the  first. 
Tiiese  exchange-values,  or  tliesc  variations  of  exchange-values,  are  similar  and 
co-ordinate  tilings  between  which  it  is  perfectly  proper  to  draw  averages. 


218         tin:    (ilKSTloN    OF     rilK    MKANS    AND    AVERAGES 

values,  iir  < if  price,-.  In  this  t'orni  tlic  prohlcin  may  hi' extended 
to  complex  eases,  eiuhracinii-  more  than  two  classes,  and  classes 
unevenly  imjiortant.  It  then  becomes  a  question  of  mathemat- 
ical averaircs,  still  with  single  weii^htinu:,  between  ]>rice  varia- 
tions— i'oy  in  tliese  cases  tlic  variations  of  the  exclinnac-\"aliies 
of  money  have  i>'enerallv  been  relei;'ate(l  to  the  rear  Ix'hind  tlie 
variations  of  prices. 

Now  we  may  find  that,  while  the  (piestictn  of  the  mean  may 
be  definitely  answerable  in  the  simple  case  posited,  yet  the  ques- 
tion of  the  averages  may  not  be  in  the  complex  cases,  except  on 
special  occasions.  It  is  plain  that  the  above  indication  of  the 
answer  to  our  general  problem  is  not  necessarily  c()m})lete.  The 
universal  solution  may  perhaj^s  l)e  yielded,  not  by  a  mathemat- 
ical average  of  the  price  variations,  but  l)y  a  comparison  of 
mathematical  averages  of  prices  at  the  two  periods  compared, 
possibly  recpiiring  the  use  of  double  weightiug.  AVe  at  least 
see  that  we  have  these  more  complex  solutions  in  reserve,  in 
case  the  more  desirable,  because  sim2)ler,  solutions,  which  first 
suggest  themselves,  shall  fail. 

II. 

§  1.  The  suggestion  that  the  general  exchangc-vahic  of 
monev  has  chantred  to  some  averau'C  of  the  variations  of  its 
many  exchange-values,  or  to  the  reci^irocal  of  some  average  of 
the  price  variations,  is  so  ])lain  that  it  has  mostly  been  acted 
uj)on  without  question.  Indeed  it  was  f  )llowcd  fi»r  more  than 
a  century  before  the  ])robl('in  itself  was  even  definitely  raised. 
And  it  was  followed  very  si)ecifically.  Among  the  three  aver- 
ages was  selected  the  arithmetic,  this  ))eing  the  simplest,  the 
easiest  to  manage,  the  most  familiar,  and  thcrcibrc  tlic  Hrst  to 
recommend  itself.  The  adoption  of  this  average  has  been  due 
to  no  considcraticju  of  its  special  proj)riety  for  the  suliject  it  is 
a])plied  to,  but  merely  to  a  general  ])reference  for  it  in  cNcry 
case.  This  is  shown  1»\'  the  wav  it  has  been  a|)plic(l.  Statis- 
ticians have  mostly  conducted  their  investigations  about  varia- 
tions in  the  exchange-value  of  moiu'v  sim|)lv  bv  noting  variations 
in    the  prices  of  commodities  ;    in   doing  which    i\\v\  lia\ c  o|tci'- 


IllsrolIV    OF     11  IK    (^XHSTION  219 

atcd  upon  the  variations  ot"  tlio  (.'Xchaiige-value.s  of  coimuoclities 
in  money,  and  not  directly  upon  the  variations  of  the  exchange- 
values  of  money  in  commodities,  but  inversely  n|)on  the  reci])- 
rocals  of  these.  It  happened  then  tliat,  dealing  merely  Avith 
prices,  the  early  workers  in  this  field  applied  the  arithmetic 
average  to  the  variations  of  prices ;  and  most  of  the  later  ones 
have  followed  suit.  Thus  they  have  unwittingly  employed  the 
most  difficult  and  least  familiar  average,  the  harmonic,  for  the 
direct  averaging  of  the  variations  of  the  particular  exchange- 
values  of  money.  Had  they  happened  to  conduct  their  en- 
quiries directly  upon  these,  as  measured  by  the  quantities  of  the 
things  purchasable  with  the  money-unit,  it  is  likely  they  would 
still  have  employed  the  arithmetic  average,  which  then  would 
be  the  harmonic  average  of  the  price  variations, — and  witli 
equally  plausible  reason,  if  a  reason  be  desired,  since  this  aver- 
age is  the  one  which  above  presented  itself  to  us  as  the  most 
specious  at  the  tirst  glance,  when  we  looked  at  the  subject  from 
this  side.  In  fact  this  course  has  actually  been  pursued.  In 
India  it  is  customary  to  report,  not  the  prices  of  corn,  but  tlie 
numbers  of  seers  purchasable  per  rupee ;  ^  and  in  averaging 
these  the  arithmetic  average  has  been  used.-  And  not  long  ago 
at  least  one  statistical  historian,  ]\I.  I'abbe  Hannauer,  presented 
his  facts  in  this  form  and  applied  to  them  the  arithmetic  aver- 
age, thus  really  using  the  harmonic  average  of  the  price  vari- 
ations.^ 

While  many  writers  were  so  engaged  in  arithmetically  aver- 
aging variations,  generally  of  prices,  with  single  weighting,  an- 
other set  of  writers  arithmetically  averaged  separately  the  prices 
at  each  of  the  periods  compared,  but,  likewise  using  single 
weighting,  on  the  assumption  that  the  same  mass-quantities  were 
produced  or  consumed  at  all  the  periods  compared.  In  doing 
so  they  employed  a  different  method;  for  the  others,  whether 
they  used  even  or  uneven  weighting,  made  no  such  assumption 

^  See  Prices  and  wages  in  India,  compiled  in  the  Statistical  Branch  of  the 
Department  of  Finance  and  Commerce,  Calcutta  1885,  (under  the  direction  of  J. 
E.  O'Conor).     Cf.  D.  Barbour,  llieory  of  bimetallism,  London  1880,  p.  121. 

-  By  Palgrave,  copying  from  Charles  C.  Prinsep,  B.  77,  pp.  378-38.3. 

'  See  B.  ,35.    Also  D'Avenel,  in  B.  117,  seems  to  have  pursued  Ww  same  course. 


22(»       rill-:  (iiKs'iiox  of    thk  mkans  and  a\'i:]!A(;ks 

al)()Ut  the  iua.<s-(juantiti('s.  Yet  none  of  tlic  writers  in  tiiese  two 
sets  seemed  to  recognize  that  they  were  })ursnin(i'  two  ditfereut 
(thongh  eonvertible)  lines  of  incjnirv  (the  ones  hailing  from  C^arli, 
the  others  from  Scrope, — for  the  followers  of  Dntot  do  not  de- 
serve further  notice).  And  we  have  seen  that  both  these  sets 
have  admitted  many  inconsistencies  in  their  methods,  consisting 
of  kinds  of  weighting  never  intended  (among  them  e\en  double 
weighting)  doing  so  because  they  carelessly  did  not  examine  what 
they  Avere  doing,  and  were  led  astray  1)y  mere  convenience. 

^  2.  The  merit  of  raising  the  question  between  the  three  aver- 
ages belongs  to  Jevons,  who  took  up  the  subject  in  1863.  Jevons 
belonged  to  the  first  of  the  above  two  sets — and  avowedly  so, 
for  Ave  have  seen  that  he  denied  the  propriety  of  averaging  prices 
at  any  one  period  alone.  He  ])osed  the  question  in  somewhat 
the  same  way  as  has  above  l)een  done,  but  very  barely.  *  At 
first  he  does  not  appear  to  have  been  acquainted  Avith  the  Avork 
of  his  predecessors,  and,  coming  to  the  subject  Avithout  prejudice, 
he  stumbled  u])on  the  ])roblem  at  the  outset,  and  at  (Mice  decided 
it  in  faA'or  of  the  geometric  mean  and  aA^-rage  (Avith  even  Aveight- 
ing),  from  which  he  was  not  repelled  by  its  difficulty,  because, 
being  a  mathematician,  he  kncAV  the  aid  to  be  derived  from  the 
use  of  logarithms.  He  Avas  very  brief  in  explaining  Avhy  he 
chose  this  ansAver,  and  the  reason  which  determined  him  is  only 
very  slightly  indicated.  It  seems  to  Iuia'c  been  that  variations 
of  prices  are  A'ariations  of  ratios,  and  the  })roper  method  of  aver- 
aging ratios  is  always  the  geometric.  ^  Within  a  year  his  posi- 
tion Avas  assaile(l  by  Professor  Laspeyres,  Avho  advocated  the 
arithmetic  average  (still  with  even  Aveighting),  and  uoav  for  the 
first  time  adA'anced  an  ai-gument  in  its  behalf.  This  argument, 
Avhich  we  shall  examine  in  detail  later,  Avas  that  the  arithmetic 
average  of  the  price  variations  marks  the  variation  in  the  sinn 
of  money  needed  to  purchase  the  same  goods  at  the  two  periods 
compared.      Jevons  \\a>   not  converted,  and  yet  he  shoAved   no 

*  lie  asked  luerely  :  If  one  thing  riseis  in  price  from  100  to  l.'iO  and  aiiotlier 
from  100  to  120,  what  is  tin-  average  rise  of  prices?  B.  22,  p.  23. 

^^  H.  22,  pp.  2.'5-24.  This  reason  was  later  more  plainly  stated  in  his  Principles 
of  urioice,  1874  ^2d  ed.,  p.  3(il),  in  a  passage  wiiich  will  later  he  quoted  (in 
Chapter  VIII.) ;  and  again  in  a  note  added  to  B.  23,  p.  12.S. 


IllsroiiV    OF    THE    QUESTION  221 

siirn  of"  (Ictectino-  tlic  error  in  Las])evres's  arjLiunK'nt.  Ho  roaf- 
tiriiK'd  his  position  the  next  year,  and  now  introdnerd  notice  of 
the  harmonic  mean  of"  ])rice  variations,  again  for  the  first  time, 
which  mean  (with  even  weightinii;),  he  pointed  out,  marks  tlie 
variation  in  the  quantity  of"  goods  the  same  sum  of"  money  evenly 
distributed  will  })urchase  at  the  two  periods.  This  average  he 
set  over  against  the  arithmetic  average,  and  not  being  able  to 
see  whv  either  should  be  jiref erred  to  the  other,  he  found  satis- 
faction in  his  geometric  average  as  lying  between  those  ecpially 
good  extreme  averages,  and  therefore  combining  the  excellences 
of  both.  ^  He  added  :  "  It  is  probable  that  each  of  these  is  right 
for  its  own  purposes  when  these  are  more  clearly  understood  in 
theory."  But  instead  of  taking  pains  to  analyze  the  subject 
further  in  order  to  reach  the  needed  clearer  understanding  of  it, 
he  contented  himself  with  the  above  reasons,  to  which  he  added 
the  following  :  "  because  it  [the  geometric  mean]  presents  facil- 
ities for  the  calculation  and  correction  of  results  by  the  continued 
use  of  logarithms," — a  worthless  reason  in  science,  and  a  foolish 
one  to  oppose  to  the  much  more  facile  arithmetic  average.  "^ 
Consequently  he  never  felt  sure  of  his  position,  and  often  s])oke 
doubtfully  about  it,  ^  and  took  refuge  in  the  thought  that  at  all 
events  he  would  be  erring  on  the  safe  side ;  for  he  was  writing 
to  prove  that  there  had  been  a  general  rise  of  prices,  and  there- 
fore he  preferred  to  underestimate  rather  than  to  run  any  risk  of 
overestimating  it.  ^ 

Following  Jevons,  the  geometric  average  has  been  adopted 
by  the  eminent  mathematical  economist,  Professor  Walras,  who, 
however,  has  found  no  better  reason  to  offer  for  it  than  Jevons's 
second  reason  (about  its  being  midway  between  the  other  two 
averages),  even  asserting  this  to  be  the  only  reason  for  employ- 

6  B.  23,  pp.  120-121. 

"  This  is  given  as  the  second  in  a  summing  up  containing  three  arguments, 
ibid.,  pp.  121-122.  The  third  is  the  first  one  above  noticed,  the  original  one.  The 
first  is  the  hiter  one  just  explained. 

*"  It  may  be  a  matter  of  opinion  which  result  is  the  truer  "  (the  geometric  or 
the  arithmetic),  B.  24,  p.  1.54.  And  he  maintained  similar  reserve  in  regard  to 
these  averages,  still  looking  upon  the  question  as  unsettled,  in  his  Monei/  and  the 
mechanism  of  exchange,  187."5,  p.  3.32. 

^  Cf.  B.  23,  p.  122,  B.  24,  p.  154.  This  has  sometimes  been  taken  for  his  prin- 
cipal reason.     It  was,  of  course,  an  afterthought. 


222         THE    (^I'KSTION    OF    THK    MKAXS    AND    AVKRAfJES 

inji'  it.'"  Conse(jU('Utly  he,  too,  has  been  weak  in  the  faith  ;  and 
he  has  inclined  more  and  more  to  prefer  the  arithmetic  average 
in  one  of  its  forms,  without  recojrnizino-  it.''  But  his  choice 
has  really  lain  only  between  the  geometric  average  with  even 
weighting  and  the  arithmetic  average  in  a  form  embracing  un- 
even weighting,  so  that  it  is  only  natural  he  should  have  pre- 
ferred the  latter.  With  this  restricted  view  he  has  not  given 
the  geometric  average  a  fair  trial,  foiling  to  perceive  that  it  also 
can  be  used  with  allowance  for  the  relative  sizes  of  the  classes.^^ 
The  cause  of  the  geometric  average  being  thus  feebly  pleaded, 
it  has  met  with  little  favor, '^  in  spite  of  the  high  authority  of 
its  ori^rinator,  who  has  even  been  treated  Avith  scant  courtesv.^* 
More  recently  Professor  Westergaard  has  advanced  for  it  tlie 
argument  which  we  have  already  examined  and  found  Avanting. 
And  occasionally  Professor  Edgeworth  has  put  in  a  good  word 
for  it.'-' 

§  ;>.  Modern  workers,  then,  have  continued  to  make  use  of 
the  arithmetic  aA'crage  of  prices,  or  of  price  variations.  They 
have  done  so  commonly  Avithout  regard  for  Laspeyres's  or  for 
anv  other  argument,  properly  so  called,  in  its  defense,'"  but,  like 
the  older  writers,  merely  because  it  is  tlie  easiest  to  execute  and 

10  B.  69,  p.  15. 

11  See  Appendix  C,  IV.  i;  1. 

12  The  geometric  average  as  used  by  Jevoiis,  with  even  weigiitiiig,  has  similarly 
been  rejected,  without  notice  of  the  possibility  of  its  being  used  with  uneven 
weighting,  by  Wicksell,  B.  139,  p.  S. 

1^  The  only  other  investigator  who  has  actually  used  it  is  Forbes.     See  B.  78. 

I*  Jevons's  use  of  the  geometric  average  was  treated  with  levity  by  Paasche,  B. 
33,  p.  .')4,  and  as  a  curiosity  by  Lehr,  B.  (58,  p.  41  n.  Marshall,  without  argument, 
considers  it  "  a  mathematical  error,  the  one  flaw  in  his  unrivalled  contributions 
to  the  theory  of  money  and  prices,"  B.  93,  p.  372  n.;  Pierson  pronounces  it  "clearly 
a  mistake,"  p.  130  in  tlie  article  referred  to  in  note  to  B.  122  ;  and  Padan  attacks 
it  savagely,  but  with  little  comprehension,  B.  141,  pp.  171-180.— Walras's  use  of 
the  geometric  average  seems  to  have  been  passed  l)y  unnoticed. 

i^'Tlie  latter  recommends  it,  with  even  weighting,  in  case  we  arc  seeking  the 
"  Determination  of  an  Index  irrespective  of  the  quantities  of  commodities;  upon 
tlic  hypothesis  that  tiiere  is  a  uumerous  grouj)  of  articles  whose  prices  vary  after 
tile  manner  of  a  perfect  market,  with  changes  affecting  the  supply  of  money,"  B. 
.")9,  pp.  280-289.  This  seems  to  refer  to  cases  when  all  prices  vary  alike,  in  which 
cases  the  weighting  is  indifferent.  But  in  these  cases  the  kind  of  average  itself  is 
also  indifferent.  In  no  other  case  do  we  want  to  seek  any  determination  "irre- 
spective of  the  quantities  of  commodities." 

"'  Laspeyres's  argument  is  reviewed,  favorably,  by  Lindsay,  B.  114,  p.  12,  and 
l)y  Mayo-Smith,  B.  137,  p.  492. 


HISTORY    OF    THE    QUESTIOX  223 

because  they  have  seen  no  prwf  of  the  superiority  of  any  other. 
Moreover  some  experiments,  by  no  means  conckisive,  show  what 
has  been  deemed  small  divergence  in  the  results  yielded  by  the 
different  methods.  Hence  it  has  seemed  like  a  dictate  of  wis- 
dom to  adoi)t  the  easiest,  so  long  as  we  do  not  know  it  to  be 
wrong.'"  But  it  is  not  an  exhibition  of  the  })roper  scientific 
spirit  to  be  content  to  remain  in  this  ignorance.^"  Beside  Pro- 
fessor Walras,  a  few  writers,  after  avowedly  rejecting  all  three 
averages,  have  lighted  upon  the  arithmetic  average  again,  in  one 
of  its  forms,  without  recognizing  it.''' 

§  4.  Consequently  also  the  harmonic  average  of  prices,  which 
is  the  most  difficult  of  all  to  manage,  and  the  one  to  which 
people  are  the  least  accustomed,  has  met  with  the  least  favor,  and, 
in  fact,  has  generally  been  altogether  ignored,  even  on  the  rare 
occasions  it  has  virtually  been  applied.  Yet,  as  we  have  seen, 
it  is  this  average  which  is  first  likely  to  appeal  to  us  when  m'c 
approach  the  subject  from  the  })oint  of  view  of  the  particular 
exchange- values  of  money  in  other  things,  measured  by  the 
quantities  of  the  other  things  the  money-unit  will  purchase  ; 
where  it  corresponds  to  the  arithmetic  average.  This  rea.son, 
which  still  manifests  a  predilection  for  the  arithmetic  average, 
is  the  one  which  led  Jevons  to  suggest  the  liarmonic  average  of 
prices.  And  it  has  since  led  one  }>romineut  economist  and 
statistician.  Professor  Messedaglia,  to  recommend  its  use  in  in- 
vestigations about  variations  in  the  exchange-value  of  money. 
This  writer  thought  he  showed  the  suitability  of  the  harmonic 

1"  Thus  its  retention  is  advised  by  Edgeworth  oil  the  principle  that  "beggars 
cannot  be  choosers,"  B.  05,  p.  386. 

•*  It  should  lie  noticed  that  in  the  usual  method  of  comparing  every  subsequent 
period  with  a  single  original  base,  the  divergences,  which  are  no  greater  in  the 
last  than  in  the  first  comparison,  are  not  of  much  consequence — and  this  whole 
method  being  wrong,  it  hardly  matters  which  average  is  used.  But  in  the  proper 
method  of  comparing  each  period  with  the  immediately  preceding  and  of  forming 
a  series  from  the  results  so  obtained,  a  very  slight  error  in  every  comparison, 
which  might  perhaps  have  a  tendency  to  work  in  the  same  direction,  caused  by 
the  use  of  a  wrong  average,  could  rapidly  accumulate  to  an  absurd  extent.  Hence 
the  selection  of  the  right  average  is  of  the  utmost  consequence.  Having  the  right 
average,  or  the  right  method  we  should  probably  make  small  mistakes  in  every 
calculation,  but  by  the  law  of  probability  these  would  fall  ubmit  equally  on 
each  side  and  would  compensate  one  another  in  the  long  run. 

13  See  Appendix  C,  Y.  H  "•  2. 


'224         TlIF,    (^rKSTION    OF    THE    MKANS    AND    AVK1!A<;ES 

average  of  ])rice  variations  for  this  ])ur|)()S('  In-  jxiintiuii-  out — 
here  like  Jevons — that  it  iiidieates  the  eommon  variation  in  tlie 
quantities  of  goods  purehasable  at  the  different  periods  ^vith  the 
same  sum  of  money  ;  for  it  thus  indicates  the  variation  in  tlie 
"  capacity  of  acquisition,"  or  purchasing  ])ower,  of  this  sum  of 
money.  On  tlie  other  hand,  here  like  Laspeyres,  he  })ointed 
out  another  purpose  served  by  the  arithmetic  average  of  jn*iee 
variations,  namely  to  indicate  the  common  variation  in  the  sums 
of  money  needed  to  purchase  the  same  goods  at  the  two  periods  ; 
only,  unlike  I^aspeyres,  he  did  not  think  this  shows  the  suit- 
ability of  this  average  for  the  previous  purpose,  for  which  it 
has  ordinarily  been  used,  as  it  indicates  rather  the  variati<in  in 
the  capacity  of  goods  to  acquire  money.  The  geometric  aver- 
age of  price- variations  he  thought  to  be  unsuitable  for  either  of 
these  purposes,  and  not  finding  any  other  purpose  for  which  it 
is  suitable,  he  considered  it,  in  our  subject,  good  for  nothing.^" 
Thus  he  has  left  the  subject  divided,  with  two  distinct  solutions, 
where  Jevons  tried  to  mediate  and  to  give  one  solution.  In  do- 
ing so  he  has  made  no  use  of  his  knowledge  that,  in  his  own 
words,  "  the  geometric  mean  .  .  .  corresponds  to  a  di/namic  con- 
cept, of  movement."  ^'  As  exchange-value  is  a  power,  or  di/- 
namis,  it  is  strange  that  his  attention  was  not  called  to  the  pe- 
culiar appearance  of  fitness  in  the  geometric  average  to  serve 
our  very  jiurpose. 

It  may  })e  added  that  a  few  writers  have  tentatively  jiut  for- 
ward the  so-called  "median  mean  "  and  the  "mean  of  greatest 
thickness."  ^^  These  are,  j)ro])er]y  speaking,  not  "means  "at 
all,  but  only  "averages,"  having  no  existence  between  two 
((uantities.  Jii  an  hypothesis  concerning  the  mean  between 
variations  of  only  two  j)rices,  as  they  have  noplace,  they  do  not 
(;all  for  attention. 

§  5.   While  Jevons  and  Ijaspeyres  were  debating  the  merits  of 

20  B.  52,  pp.  38-40. 

21  Ibid.,  p.  30. 

22  The  median  mean  is  especially  favored  by  Edgeworth.  He  recommends  it 
when  we  are  seeking  the  "  Determination  of  an  Index  based  upon  quantities  of 
commodities  ;  upon  the  hypothesis  that  a  eommon  cause  has  produced  a  general 
variation  of  prices,"  B.  5!l,  pp.  2S9-203.  See  also  B.  Gl,  pp.  360-303,  and  B.  65, 
[).  74. 


iiis-i'(»i:v   (»i-    Till-;  (iiKSTioN  '2'2^ 

the  geometric,  nritlimetic  jiiid  hannonic  means  apj^lied  to  price 
variations  "with  even  weigliting',  the  economist  Hoscher  submit- 
ted the  question  for  (k'cision  to  the  mat]i(>mati('ian  Drobiscl'. 
Drobisch  (h'cided  it  bv  sunnnarily  rejecting  all  three  of  the  means 
— although  what  he  really  did  was  to  reject  them  with  even 
weighting  for  conditions  demanding  uni'ven  weighting.  He  also 
did  more.  He  rejected  the  averaging  of  ])ricc  variations  with 
single  weighting,  and  rejected  the  use  of  the  same  mass-quanti- 
ties at  both  periods  in  averaging  the  ])i'ices  at  each  period.  Tn 
their  })lace  lie  introduced  the  method  of  double  weighting — with 
the  arithmetic  average,  and  with  what  we  have  seen  to  be  a  mis- 
taken method  of  selecting  the  mass-units.  Elxcept  for  a  vigor- 
ous reply  by  Laspeyres,  this  new  method  met  with  little  notice, 
and  was  almost  unknown  outside  of  Germany.  In  Germany, 
however,  an  improvement  upon  it  was  later  made  by  Professor 
lichr.  And  in  England  another  economist,  Professor  Nichol- 
son, apparently  without  knowledge  either  of  Drobisch  or  of  Lehr, 
invented  a  method  very  similar — in  fact,  in  some  aspects  quite 
similar — to  Drobisch's.  These  three,  Drobisch,  Lehr  and 
Nicholson,  form  another  distinct  line  of  theorists, — diiferentiated 
from  the  rest,  however,  rather  on  the  subject  of  weighting  than 
on  the  subject  of  the  averages. 

The  position  of  these,  the  most  difficult,  and  intended  to  meet 
the  most  difficult  cases,  we  may  leave  to  the  last  in  our  attempt 
to  solve  anew  the  problem  above  ])Osed — a  problem  raised  more 
than  a  third  of  a  century  ago  and  not  yet  settled.  We  may 
])roceed  circums])ectly,  beginning  with  the  simplest  and  most 
easily  suggested  answers  to  the  simplest  form  of  the  jiroblem.  The 
simplest  and  first  suggested  answer  is  that  of  a  mean  or  average 
of  the  price  variations,  with  single  weighting.  This  we  must 
examine  first.  If  it  fails  us, — or  when  it  fails  us,  for  it  may 
])erhaps  suffice  for  the  very  simplest  form  of  the  problem  (with 
two  classes  and  even  weighting), — we  must  then  })ass  on  to  seek 
other  solutions  on  other  lines — either  Avith  supposedly  perma- 
nent mass-quautities  or  with  double  weighting. 

15 


("IIAITKK    VII. 

BHIP]F   COMPARISON    OF   TIIK    MKAXS. 

T. 

^  1.  Before  examinino;  the  aruiinieiits  ])r()per  for  the  diiferent 
mathematieal  mean^  of  the  variations,  it  will  be  well  to  com])are 
these  means  with  one  another.  The  ('t>ni])arison  will  not  only 
make  us  better  aec^uainted  with  them,  but  may  j^erhaps  even 
give  some  indication  of  the  superiority  of  one  of  them  by  dis- 
closing shortcomings  in  the  other  two,  and  so  aid  our  under- 
standing of  the  arguments.  For  this  ])urpose  it  will  be  advisable 
to  put  our  siin])le  ])roblem  in  another  f  )rm.  We  may  make  use 
of  another  ])ieee  of  information  already  possessed.  This  is  that 
if  one  article  rises  or  falls,  its  variation  may  be  eomijensated  by 
an  oj)posite  fall  or  rise  of  another  article.  As  yet  we  do  not 
know  how  great  must  be  the  opposite  variation.  Here  also  is  a 
Proposition  lacking,  which  we  wish  to  find.  The  seeking  after 
it  is  the  same  operation  as  the  seeking  after  the  mis-iiig  Prop- 
osition referred  to  in  the  last  Chaptei",  as  will  l)e  seen  in  a 
moment.  Keeping  to  our  simplest  suj)positional  case,  we  may 
ask  this  question  : — 

Among  three  classes  of  things,  two  being  e<|ually  important, 
whose  units,  AI,  A  and  ]>,  are  at  first  eipiivalent,  if  one  of  them, 
A,  rises  in  exchange-value  in  [MJ  by  one-half,  so  that  its  |)rice, 
from  being  l.OO,  becomes  l.oO,  and  so  that  M,  from  purciiasing 
A,  comes  to  purcihase  only  f^  A,  how  imich  of  B  nuist  M  |)ur- 
chasc  in  order  that  its  exchange-value  in  [A]  and  [B]  together 
shall  remain  as  it  was  before,  and  to  what  price,  from  1.00,  will 
the  price  of  B  then  fall? 

Her(!  the  new  exchange-valnc  of  M  in  [A]  an<I  the  new  ex- 
change-vahie  of  M  in  []>]  will  have  the  old  exchange-values  of 

226 


ANOTirER    VOK^r    or    TTTF.    I'RORT,i:>r  227 

M  in  [A]  and  in  [B],  whicli  were  units,  between  them,  and 
therefore  possibly  as  a  mean  of  some  sort ;  and  consecjuently  the 
new  price  of  A  and  the  new  j)rico  of  B  will  have  their  old  prices, 
1.00,  between  them  as  a  mean  of  a  corresponding  sort.  Only 
instead  of  seeking  the  mean  between  two  given  extremes,  we  are 
seeking  the  absent  extreme  when  one  extreme  and  the  mean  are 
given.  Hence  this  problem  is  the  same  as  the  problem  set  in 
the  last  Chapter. 

In  this  problem  we  shall  for  the  present  assume  that  the  classes 
are  equally  large  or  important,  not  at  either  of  the  periods  alone, 
but  at  each  of  the  periods,  or  somehow  over  both  the  periods. 
But  we  shall  attempt  to  leave  the  (juestion  of  weighting  out  of 
sight  so  far  as  possible.  We  have  already  treated  the  question 
of  weighting  by  itself,  as  far  as  we  could  ;  and  now  we  shall 
try  to  treat  the  question  (jf  the  averages  by  itself,  as  far  as  we 
can.  We  shall,  however,  find,  as  we  proceed,  that  we  can  not 
settle  this  question  any  more  than  that  question  separately.  We 
shall  then  have  to  combine  the  two  into  one  question  about  the 
variation  of  the  general  exchange-value  of  money  under  given 
conditions  of  variations  both  of  prices  and  of  mass-quantities. 
This  course  is  adopted  because  it  repeats  the  usual  way  the  sul)- 
ject  has  been  approached,  although  only  three  writers  have  as  yet 
carried  it  to  the  end.  In  pursuing  it  all  opinions  hitherto  ad- 
vanced may  be  reviewed. 

§  2.  The  problem  in  its  new  form,  may,  for  the  exchange- 
values  of  M,  be  formulated  in  our  three  formulse,  in  the  order 
with  which  we  are  familiar,  as  follows  : 

j/,,  =  i(i-fi)  =ii4.  =  i(i;  +  ^'), 


1/1    IN""'"!/!     i^y 


Mo,  =  V\l  =Mo,  =  \ 


For  these  formulae  to  be  carried  out,  that  is,  for  ^^,,,  to  be  equal 


•22<S  i'.i;ii;i"  comi'aimson  or  the  >h:axs 

to  J/^ij  ^^'P  ^\■allt  tho  resultants  of  tlie  fii>uros  in  the  seeond  halves 
to  ecjual  tliose  in  the  tirst  iialves,  wliich  are  all  units.  AVe  see 
that  this  will  he  the  ease  in  the  first  tnrinula  if  h'  =  \l,  in  the 
second  if  h'  =  2,  in  the  third  if  6'  =  1  J.  Thus  if  we  later  iind 
any  one  of  these  figures  to  be  the  right  answer,  Ave  shall  kno>v 
that  its  formula  is  the  proper  one,  in  similar  cases,  for  averag- 
ing particular  exchange-values  of  M  in  all  other  things  at  two 
periods  for  the  purpose  of  comparing  its  exchange-value  in  all 
other  things  at  those  periods,  as  it  alon(>  gives  the  right  answer 
in  this  simple  ease. 

The  corresj)onding  formula  for  the  prices  are,  in   the  same 
order  : 

1/  1       2^ 

2  \  l.(H)       1.00 


P_  ==     },(1.()0  +  1.00)   =  P,  =    ^(1.50  -f  /9'), 
\\=    I    1.00x1.00    =1\,  =    1   1.O0/3'. 

And  similarly  we  want  the  results  in  the  second  halves  to  equal 
unity,  which  will  be  the  case  in  the  first  formula  if  ,^'  =  .75,  in 
the  second  if  ^9'  =  ,50,  in  the  third  if  ^5'  =  .()(>§  ;  which  are  the 
reciprocals  of  the  ])receding  answers  for  b' ,  in  the  same  order. 
That  one  of  these  formulae  for  averaging  prices,  if  any,  will  be 
the  pro])er  one,  in  which  the  answer  for  ^9'  is  the  reciprocal  of 
the  pro])er  luiswer  for  h' .  * 

An  advantage  in  re-stating  the  question  in  this  form  is  that 
the  diversity  of  the  proffered  extremes  is  greater  than  the  di- 
versity of  the  ])r()ffered  means.  Also  it  brings  in  the  idea  of 
balancing,  in  which  the  idea  of  equality  of  influcucc  is  plainer 
than  in  the  case  of  conjoint  action.  The  review  and  comparison 
of  the  suggested  answers  will  now  disclose  interesting  relations. 
And  two  of  the  answers  will  manifest  unmistakable  signs  of 
falsity. 

II. 

§  1.  Only  the  |)iirchasal)lc  ({uantities  of  the  articles  being 
considered  (which  are  dircctlv  accordinsi;  to  the  exchaniic-values 


THE    COMPARISON  2211 

of  M  in  tlu'iu),  it  is  easy  to  answer  tliat  if  M  will  purchase  only 
I  as  niueli  of  [A]  as  before,  it  should  purchase  ^j  more  of  [B] 
in  oi-der  to  make  uj)  for  the  .V  lost  on  [A].  Here  the  new  ([uan- 
tities  purchasable  with  M  (and  the  new  exchantjc-values  of  M 
in  [A]  and  [Bj )  surround  the  old  (juantities  (and  exchange- 
values),  which  were  units,  so  as  to  hold  them  in  the  arithmetic 
mean,  the  progression  being  |,  1,  IJ.  The  new  prices,  then, 
are  1.50  for  A  and  .75  for  B,  which  hold  the  old  jn'iees,  1.00, 
as  their  harmonic  mean,  in  the  })rogression  1.50,  1.00,  .75,  The 
|)roj)er  formulae,  if  this  answer  be  correct,  are 


-=2(^  +  4)    ""^^    P^  =  l7j 

2V1.5 


)0  "^  .75 

the  result  in  each  of  these  being  unity. 

Only  the  prices  being  considered,  it  is  easy  to  answer  that  if 
A  has  risen  to  1.50,  B  should  fall  to  .50,  so  that  the  increase  of 
money  needed  to  ])urchase  A  shall  be  made  up  for  by  the  de- 
crease of  money  needed  to  purchase  B.  Here  the  new  prices 
have  the  old  ]>rices  as  their  arithmetic  mean,  the  progression 
being  1.50,  1.00,  .50.  On  examining  the  new  quantities  pur- 
chasable with  M,  we  find  that  they  are  f  A  and  2  B,  which 
have  the  old  quantities,  which  were  units,  between  them  in  the 
harmonic  mean,  according  to  the  progression  §,  1,  2.  The 
proper  formulae  for  this  answer  are 

-V,„  =  ,-7-r~-j,       and     P,  =  1  (1.50  +  .5(1). 

Both  the  ([uantities  and  the  prices  being  considered,  it  is  pos- 
sible to  treat  them  uniformly  by  answering  that  if  the  price  of 
A  rises  to  |  times  its  former  position,  the  price  of  B  should  fall 
to  I  of  its  former  position,  and  that  if  the  quantity  of  [A]  pur- 
chasable with  ]M  falls  to  5  of  its  former  amount,  the  quantity  of 
[B]  purchasable  with  M  should  rise  |  times  its  former  amount, 
— whereby  are  |)roduced  the  same  variations  in  the  exchange- 
values  of  A  in   [M]  and  of  B  in  []M]  as  in  the  exchange-values 


2:50 


I'.IMKF    ( OMI'AKISOX    OF    THE    MEANS 


of  M  in  [BJ  and  of.Miii  [A].  Here  botli  tluMicw  pi-iccs  and 
till' nt'W  quiintitics  have  their  old  [)rice.s  and  their  old  (juantities, 
which  were  units,  between  them  us  geometric  means,  in  the  pro- 
gressions |,  1,  1,  and  |,  1,  |.     The  ])r()per  formulae  are 


J/...  =  I 


U      and      V.,=  I    1.50  x  Mf,. 


§  2.  Thus,  so  to  speak,  the  harmonic  and  arithmetic  aver- 
ages are  locked  in  an  inseparable  embrace,  but  the  geometric 
stands  by  itself,  self-sufficient.  And  both  its  answers  are  mid- 
Avay  between  those  of  the  others.  The  ])eculiarities  of  these 
tiiree  methods  of  answering  may  best  be  shown  by  drawing  up 
a  tabl(»  of  several  supposed  variations  in  price  of  A  and  of  the 
counterbalancing  variations  in  price  of  B  according  to  the  three 
methods,  and  paralleling  them  with  the  variations  in  the  (pian- 
tities  purchasable  with  M. 


Counterbalancing         1 

0 

Counterbalancing  new 

is    . 

0 

new  ] 

)rice.s  of  B  accord- 

(juautities  of  B  accord- 

lug 

to  progressions  : 

R  0 

•^s? 

ing  to  progressions  : 

•T3  *S 

.H-s 

|.s 

I'g 

g  o 

1                1 

■2< 

5.2 
"C  » 

i  ^1 

1  u 

0 

Har- 
monic. 

Geo- 
metric. 

A3, 

1.25 

1.00 

.75 

.80 

.83^ 

1.8 

1.333^     1.25       1.2 

1.331 

1.00 

.661 

.75 

.80 

1.75 

1.5        1.33'     1.25 

l.oO 

1.00 

.50 

.661 

.75 

1  .m\ 

2            1.5         1.33^ 

1.75 

1.00 

.25 

.57} 

.70 

1  -571 

4            1.75       1.42f 

1.90 

1.00 

.10 

R9I2 

•"-'IS 

.67f 

.b1\l 

10          1.9         1.47iV 

2.00 

1.00 

0 

.50 

.661 

.5 

00        2            1.5 

2.50 

l.UO 

.40 

.62.^ 

.4 

2.5        1.6 

3.00 

1.00 

.331 

.60" 

■33  J 

3            1.66| 

4.00 

1.0(1 

.25 

.571 

.25 

4            1.75 

5.00 

1.00 

.20 

.551 

.2 

.■)            l.S 

10.00 

1.0(1 

.10 

■52B- 

.1 

10          1.9 

100.00 

1.00 

.01 

.50^0^ 

.01 

100        1.99 

00 

1.00 

0 

.50 

\     0 

<x> 

•) 

On  lookmg  over  this  table  we  see  that  as  the  pi-iee  of  A  is 
supposed  to  rise  and  the  (piantity  of  [A]  purchasable  Avith  M 
to  fall,  the  cotmterbalaneing  fall  of  the  ])rice  of  B  according  to 
the  arithmetic  progression  is  so  i-apid,  and  also  the  correspond- 
ing rise  of  the  quantities  of  [B]  ])urchasal)le  w  itii  .M  at  these 
prices,  which  (|iiaiilities  are  according  to  the  harmonic  progres- 
si(m,  that  when  the  |)rice  of  A  reaches  2.00  and  thereafter  there 
is  n(»  connterbalancing  price  of  B,  which  would   have  to  be  zero 


rill-;   (OMi'AKisoN 


or  loss,  and  no  eounterbalanciiiii'  (|uantity  of  I)  |)iir('hasal)U'  w  itii 
M,  as  it  would  have  to  he  int'niitc  or  hccoinc  a  iicti'ative  (|naii- 
titv,  wliicli  here  is  incaniiiiilcss.  And  corrcspondinii^ly,  if"  wo 
consider  til 0  fall  in  the  j)rico  of  B  and  ask  for  the  coinitorhalanc- 
ing  rise  in  tlio  price  of  A  according  to  the  harmonic;  progression, 
with  the  quantities  in  arithmetic  pn^gession,  we  find  this  rise  so 
rapid  as  to  reach  infinity  when  the  ])rice  of  B  falls  to  .oO,  and 
thereafter  it  would  he  a  negative  ([uantity,  which  again  is  meaning- 
less here.  In  practice,  however,  these  difficulties  would  not  al- 
wavs  he  insuperable,  as  we  should  then  he  dealing  with  many  ar- 
ticles, and  the  comjiensation  might  be  obtained  by  distributing  the 
counterbalancing  change  over  two  or  more  of  them.  Thus  in  the 
case  of  the  arithmetic  solution  of  prices,  a  fall  of  prices  of  100  per 
cent,  (^r  greater  could  be  obtained  bv  dividini>'  it  into  several  lots. 
Yet  a  rise  of  1(^0  per  cent,  on  an  average  over  half  the  numbfir 
of  articles  could  not  be  counterbalanced  ])y  any  fall  of  the  others. 
In  the  case  of  the  harmonic  solution  of  ]MMces,  the  rise  to  infin- 
ity would  not  be  rocpiired  if  two  or  more  articles  rose  in  price, 
because  a  whole  or  greater  (juantity  purchasable  with  M  may 
be  lost  by  being  distributed  over  two  or  more  articles  without 
any  considerable  rise  of  their  prices.  For  instance,  the  fall  of  ] > 
to  .50  can  be  compensated  by  the  rise  of  two  articles  to  2.00. 
Yet  again  the  fall  of  half  the  number  on  an  average  to  less  than 
.50  could  not  be  counterbalanced  by  any  rise  of  the  others. 
These  great  changes  of  prices  on  the  part  of  a  majority  of  com- 
modities are  not  to  be  expected  unless  money  1)0  varying  in  its 
general  exchange-value,  and  so  the  eom])ensation  may  not  be 
practically  needed.  Yet  theory  ^\■ould  seem  to  recpiire  its  possi- 
bility always.  The  geometrical  method  satisfies  this  theoretical 
re(|uireraent.  For  every  rise  or  fall  of  one  article  it  ]irovides  a 
counterbalancing  fall  or  rise  of  another. 

>^  '•].  Tiiis  tendency  of  the  arithmetic  and  harmonic  solutions 
to  run  into  the  ground  or  to  fiy  into  the  air  by  their  excessive 
demands  is  clear  indication  of  their  falsity.  Frror  is  often  made 
[)atent  by  examining  extreme  cases.  It  is  so  here.  Take  the 
case  when  A  rises  in  ])rice  to  l.!M).  You  can  then  1)U\'  with 
one   money-unit   ten  nineteenths   as   much  of   [A]  as  you  cnuld 


2.)2  r.IMKF    (O.MI'AIMSON    dl'     I'lll-:    MEANS 

bctoiv,  or  a  triHo  over  luili'.  I'lic  arithmetic  answer  says  you 
ouiiht  then  to  hny  of  [B]  ten  times  as  much  as  before.  Thus, 
losinii'  nine  nineteenths  or  not  ([uite  lialf  on  [A],  you  uain  nine 
wholes  on  [B]  •  The  excessiveness  is  evident.  Your  money 
has  appreciated.  On  the  other  hand  consider  the  tidi  of  the 
quantity  of  [B]  to  .")-y|.  With  your  money-unit  you  can  now 
get  nine  tenths  more  of  [B]  than  before,  or  not  quite  double. 
Yet  the  harmonic  method  of  calculatin<>'  the  terms  of  prices  would 
re(|uire  the  price  of  A  to  rise  to  1  (».(»(»,  which  will  enable  you 
with  your  money-unit  to  get  only  one  tenth  so  much  of  [A]  as 
formerly.  Again  the  excessiveness  is  evident.  Yoiir  money 
has  depreciated.  The  geometric  method  commits  neither  of  these 
excesses.  In  the  former  case  it  does  not  require  the  price  of  B 
to  fall  so  low  as  did  the  arithmetic — only  to  .oOyl  instead  of  t() 
.10;  and  it  enables  you  then  to  get  nineteen  tenths  instead  of 
ten  wholes,  so  that,  losing  nine  nincteentlis  on  [.\]  ,  you  gain 
nine  tenths  on  [B]  instead  of  nine  wholes.  There  is  no  a])pear- 
ance  here  of  a])preciation.  In  tlu'  latter  case,  this  method  does 
not  require  the  ])riee  of  A  to  rise  so  high  as  did  the  harmonic — 
onlv  to  l.iM)  instead  of  to  10.00  ;  and  it  enables  you  then  to  get 
w'itli  your  mouey-imit  ten  nineteenths  instead  of  only  one  tenth 
of  [A],  so  that,  gaining  nine  tenths  on  [B],  or  not  (juite  as 
much  again,  you  lose  nine  nineteenths  on  [A]  instead  of  nine 
tenths,  or  nearly  the  whole.  Neither  is  there  lui'e  any  ajipear- 
ance  of  de|)reciation.  There  is  thus  an  ap])earance  that  the 
geometric  mean  may  have  rendered  your  money  stable  in  general 
exchange-value,  since  it  has  removed  the  appearance  both  of 
appreciation  and  of  de[)reciation. 

As  vet,  however,  we  are  but  slightly  atlvanced  beyond  .levons 
in  one  of  his  moods.  Jevons  acknowledged  himself  at  a  loss 
between  tlie  opposite  merits,  as  he  adniitte(l  them,  of  the  arith- 
metic and  harmonic  methods  of  averaging  prices,  and  rested  con- 
tent with  a  conq)romise  on  the  intermediate  geometric  method. 
We  see  the  o])posite  failings  of  tjiose  methods,  aii<l  welcome  the 
geometric  method  for  a\'oiding  each  extreme,  and  so  suggesting 
at  least  the  possibilit\"  oi"  its  being  the  correct  one — under  the 
assumed  con<litions  at  lea,-.t. 


CHAPTER    VIII. 

THE  GEXP:RAL  ARCiUMENT  FOR  THE  GEOMETRIC  MEAN. 


;^  1 .  The  problem  before  us  ha.s  two  aspects.  The  one  is  the 
point  of  view  of  prices,  the  other  the  point  of  view  of  the  ex- 
clianoe- values  of  money  in  the  goods  priced.  Of  two  equally 
important  classes  of  goods  the  rise  of  the  price  of  the  one  is  to 
Ix-  compensated  by  a  fall  of  the  price  of  the  other ;  and  the  rise 
of  the  exchange- value  of  money  in  the  latter  is  to  be  compen- 
sated by  a  fall  of  the  exchange-value  of  money  in  the  former. 
The  second  is  really  our  niain  problem;  ])ut  its  place  may  be 
taken  by  the  first,  for  the  sake  of  convenience.  No  argument, 
however,  is  good  showing  the  superiority  of  one  kind  of  com- 
pensation in  the  problem  of  prices,  unless  it  can  show"  also  the 
superiority  of  the  kind  of  compensation  thereby  involved  in  the 
problem  of  the  exchange-value  of  money.  In  order,  then,  to  l)e 
able  to  apply  the  question  of  compensation  and  of  averages  to 
our  double-faced  problem,  avc  must  understand  something  of  the 
nature  of  compensation  in  general,  and  of  the  nature  of  averages 
in  reference  to  the  kinds  of  subjects  for  wliich  they  are  generally 
suitable. 

In  the  question  of  coitijx'iisatory  variations,  the  following 
proj)osition  at  once  strikes  us  as  evident : — A  compeimdorij  cdria- 
tion  in  the  one  direction  must  equal  the  variation  in  the  other  direc- 
tion li'hich  it  is  to  compensate. 

We  generally  treat  variations  by  percentage,  and  we  have  an 
habitually  estal)lished  manner  of  reckoning  percentages.  We 
treat  an  increase  or  rise  on  the  one  hand,  and  on  the  other  a  de- 
crease or  fall,  as  variations  from  one  common  point  or  position, 
or  whole,  arbitrarily  cliosen    as   that   Avhich  exists    at   the  com- 

238 


284    (jp^NKHAi,  .\i!(;rMi:NT  Foi;    iiik  (jkomkikic  mkan 

moiicenu'Ut.  The  ])(.'rc'cnt;iu('  ot'tlic  rise  i>  the  ratio  of  the  later 
excess  beyond  the  original  whole  to  that  whole,  multi])Hed  by 
one  hundred  ;  and  tlie  |)ereentag'e  of  the  t'all  is  tlie  I'atio  of  the 
later  detieieney  below  the  oriyinal  whole  to  that  whole,  niulti- 
|)lied  by  one  hundred.  Hereupon  we  are  inclined  to  maintain — 
and  this  is,  in  almost  all  eases,  the  fundamental  idea  in  thearffll- 
ment  for  the  arithmetic  average  of  variations — that  ((  rompca- 
.s( do  1-1/ fall  .should  always  have  tit e  .same  perceiif age  a.s  flie  ri.sc  it  is 
to  compensate,  and  reversely.  But  the  above  is  not  the  only  Avay 
of  reckoning-  percentages. 

Of  reckoning  percentages  there  are  three  diiferent  \vays — or 
better,  two  distinct  ways,  and  a  third  constructed  by  nniting 
them;  wherefore  there  are  three  diiferent  ways  of  obtaining 
sameness  of  percentage  in  compensatory  variations.  Conse- 
quently the  argument  for  the  arithmetic  average  has  no  validity 
unless  it  can  be  proved  that  the  ordinary  w^ ay  of  reckoning  ])er- 
centages  is  the  proper  way  of  reckoning  them  in  the  s])ecial  case 
in  question.  The  ordinary  w^ay  of  reckoning  ])crceutages,  or  of 
measuring  variations,  is  the  way  we  shall  always  continue'  to 
pursue,  because  of  its  greater  convenience.  But  unless  it  be 
the  proper  way  for  indicating  the  equality  of  variations  in  the 
])rol)lem  before  us,  we  shall  have  to  adaj)t  it  to  our  subject  by 
ada])ting  whatever  turns  out  to  be  the  proper  method  of  meas- 
uring the  variations  to  this  convenient  method  of  measuring 
them. 

AVe  must  therefore  examine  the  three  ways  in  which  we  can 
conceive  of  equality  in  conq)ensatoi-y  variations. 

§  2.  A  being  supposed  to  rise  in  S(tme  attribute  to  a  certain 
extent,  ecjuality  to  that  rise  in  the  fall  of  ]5  in  the  same  attribute 
may  be  represented  in  the  following  three  wavs  :  (1 )  B  may 
foil  so  that  the  |)oint  to  which  it  falls  is  as  nnich  below  the  point 
from  wiiich  it  fidls  as  the  point  to  which  A  lias  risen  is  al)ove 
the  point  from  which  A  has  risen  ;  (2j  B  may  fall  so  that  the 
point  from  which  it  falls  is  as  much  above  the  ]>oint  to  which  it 
falls  as  the  point  from  w  hich  A  has  risen  is  below  the  point  to 
which  A  has  risen  ;  (."5)  B  may  fall  so  that  either  the  point  to 
which  it  falls  is  as  nnich  below  the  point  from  which  it    fidls  as 


KQUAI>ITY    I.\    OPPOSITE    VAKIATIOXS  23-"> 

the  point  from  whicli  A  has  risen  is  below  tlic  [xiint  to  which  A 
has  risen,  or  the  ])oint  from  which  it  falls  is  as  Jiiucli  above  the 
point  to  \vhieh  it  falls  as  the  point  to  Avhieh  Abas  risen  is  above 
the  point  from  wliieh  A  has  risen.  In  the  first  we  nieasni'e 
the  variations,  or  their  percenta^-es,  in  the  usual  way,  in  what 
the  thinii's  vary  from.  In  the  otiier  two  we  depart  from  the 
usual  method.  In  the  second  we  measure  the  percentages  in 
what  the  thintis  vary  to.  In  the  third  we  measure  the  varia- 
tions either  in  what  the  one  thing  rises  to  and  the  other  falls 
from,  or  in  what  the  former  rises  f/-o//i  and  the  latter  falls  to. 
In  the  first  two  we  make  each  measurement  in  an  opposite  di- 
rection— in  the  first  from  the  start,  at  the  mean,  to  the  ex- 
tremes ;  in  the  second  from  the  extremes  at  the  starting  points 
to  the  mean, — the  former  being  centrifugal,  so  to  speak,  and 
the  latter  centripetal.  In  the  tliird  we  make  both  measure- 
ments in  the  same  direction,  either  both  u])wards  or  botli  down- 
wards, the  starting  point  of  the  one  being  at  the  mean  and  tliat 
of  the  other  at  an  extreme,  or  that  of  the  latter  at  the  mean 
and  that  of  the  former  at  the  other  extreme. 

When  the  variations  in  which  e({uality  is  obtained  in  these 
three  ways  are  all  measured  in  the  usual  way,  the  first  are  vari- 
ations from  the  mean  to  the  simph' '  arithmetic  extremes  around 
it.  Thus,  for  exam})le,  A  rising  from  1.00  to  l.oO  l^y  ~){)  per 
cent,  (reckoned  in  1.00),  B  falls  from  1.00  to  .50  by  50  percent, 
(also  reckoned  in  1.00).  They  may  therefore  be  called  simple 
arithmetic  variations.  The  second  are  variations  from  the  mean 
to  the  simple  harmonic  extremes  around  it.  Thus,  A  rising 
frcmi  1.00  to  1.50  so  that  1.00  is  ;i3i  per  cent,  below  1.50 
(reckoned  in  1.50),  B  falls  from  1.00  to  .75  so  that  1.00  is  :]:'>^ 
per  cent,  above  .75  (reckoned  in  .75).  They  may  therefore  be 
called  simple  harmonic  variations.  The  third  are  variations  from 
the  mean  to  the  simple  geometric  extremes  around  it.  Thus,  A 
rising  from  1.00  to  1.50  by  50  ])er  cent.  aV)ove  1.00  (reckoned 
in  1.00),  B  falls  from  1.00  to  .(j(v|  so  that  1.00  is  50  per  cent, 
above  .66f  (reckoned  in  .G6f) ;  or,  A  rising  so  that  1.00  is  3.")^ 
])er  cent,  below   1.50  (reckoned   in    1.50),  B  fall-  fVoin  1.00  to 

^  /.  e.,  with  even  weighting,  only  two  fiirures  being  ii-^i-d. 


'2.><)      GENERAL    Al!(;r.MENT    FOR    THE    (iEoMETRIC    ^fEAN 

.()0|  f^o  that  .GG|  h'S'.i^  }X'r  cent,  below  l.(»»)  (ivckoned  in  1.00). 
They  may  therefore  be  called  simple  geometric  variatlon.'i.'^ 

Thus  when  we  use  in  all  tlie  cases  the  ordinary  method  of 
reckoning  the  ]>eroentages,  the  equality  exists  only  in  the  first 
one.  Yet  this  eciualitv  is  no  more  real  than  the  equality  which 
appears  in  tlu'  other  wijys  of  reckoning  the  percentages. 

§  .3.  The  equality  manifested  by  the  arithmetic  variations 
seems  to  be  of  a  peculiar  nature.  This  is  ecpiality  of  ditt'erence, 
()r  of  distance  traversed,  l.oO  and  ,50  being  equidistant,  so  to 
speak,  from  1.00,  wherefore  A  in  rising  from  1.00  to  l.")0  and 
B  in  falling  from  1,00  to  JA)  have  traversed  equal  distances. 
This  pr<>])erty  may  l)e  thought,  at  first  sight,  to  belong  only  to 
the  arithmetic  variations.  It  does  so  belong  only  to  them  if  the 
variations  must  l)e  conceived  as  starting  from  1,00,  or  from  some 
conmion  figure,  that  is,  from  the  mean.  But  there  is  no  reason, 
except  mere  convenience,  why  in  all  cases  the  variations  must 
be  so  conceived.  If  they  are  conceived  as  starting  from  other 
figures,  this  property  of  traversing  e(|ual  distances  may  belong- 
to  the  other  variations. 

Thus  if  A  rises  from  1.00  to  l.oO  and  B  falls  from  2.00  to 
l.oO,  A  has  risen  by  oO  per  cent,  and  B  has  fallen  by  25  per 
cent.  Evidently  the  variation,  merely  as  a  variation,  of  |  B 
from  1.00  to  .75  is  the  same  variation  as  that  oi'  B  from  2.00  to 
1,50,  Here  we  have  harmonic  \ariations.  Yet  the  distances 
traversed  are  equal. 

And  if  A  rises  from  1.00  to  1.50  and  B  falls  from  1.50  to 
1.00,  or,  B  falling  from  1.00  to  .m^,  if  A  rises  from  .6()|  to 
1.00,  A  has  risen  by  50  pei-  cent,  and  B  has  fallen  by  »"J3^  per 
cent.,  which  are  geometric  variations;  for  A  has  risen  to  be  50 
per  cent,  higher  than  it  was,  and  B  was  50  per  cent;  higher  than 
it  has  come  to  be.  But  A  and  B  have  each  traversed  an  ccjual 
distance — in  fact,  so  to  speak,  the  same  road,  only  in  opposite 
directions,  so  that  in  this  case  sameness  is  added  to  equality. 

These  pro])erties  are  universal.  All  variations  from  the  same 
starting  |)oint  to  e(|ual  distances  on  oj)posite  sides,  above  and 
below,  are  arithmetic   variations  (variati(ms   to  arithmetic   ex- 

-  The  univcrsulity  of  these  relations  is  demonstrated  in  Appendix  H,  I. 


IX^rALITV     IN    OPPOSITE    VAJilATloNS  'I'M 

tremos) ;  and  all  aritlimetio  variations  may  he  rcdnced  to  snch. 
All  variations  from  cqnal  distances  on  opposite  sides,  above  and 
below,  to  the  same  ending  point  are  harmonic  variations  (vari- 
ations to  harmonic  extremes,  when  measnred  in  the  usnal  AvayJ ; 
and  all  harmonic  variations  may  be  reduced  to  such.  All  vari- 
ations over  the  same  distances,  between  the  same  extremes,  in 
opposite  directions,  ujnvards  and  downwards,  are  geometric^ 
variations  (variations  to  geometric  extremes,  when  measured  in 
the  usual  way)  ;  and  all  simple  geometric  variations  may  be  re- 
duced to  such.* 

§  4.  Thus  in  all  these  variations  we  find  two  kinds  of  equal- 
ity :  equality  of  projwrtion,  and  equality  of  distance  traversed, 
or  of  difference.  These  may  be  represented  algebraically  as 
follows.  Let  rt,  and  a,,  represent  the  figures  at  which  A  stands 
at  the  beginning  and  at  the  end  of  its  variation,  and  h^  and  6^ 
the  similar  figures  for  the  variation  of  B.     Then  in  the  arith- 

•     •                 1          '^'■2  —  '^'l        ^^i  ~^'-  ^  -p  1 

metic  variations  Ave  have =  --.-    ",  anaitrt,=  o,,  ((.,—  ((, 

ft,,  —  (I       6,  —  6.,  ^ , 

=  b,  —  h,  ;  in  the  harmonic  variations,    '  =  — -. — ".and  it 

(I.J  o^ 

a„  =  6„  (I,  —  o,  =  h,  —  A.,  :  and  in  the  geometric  variations,  both 

a  —  a       b  —  b., 

~ =  -^ — "   and  II  a,  =  b.„  «.,  —  a,  =  Ik  —  f/.,  (whence  also 

a.j  b.,  '         ~     ^         '         ^ 

fi.j  =  o,),  and  —  -        =  — .      ",  and  it  <i.,  =  o,,  c^,  —  a,  =  '>i~'>z 


«., 


b 


(whence  also  a^  =  b.,):^ 

When  one  or  both  of  these  kinds  of  equality  are  present,  as 
they  may  appear  in  three  different  ways,  it  may  be  well  to  give 
to  these  distinctive  a}»pel]ations.  Thus  when  we  have  the 
equality  which  ap])ears  in  arithmetic  variations,  we  may  name  it 
arithmetic  equality  of  variations,  or  of  percentages.  AVhen  we 
have  the  equality  which  appears  in  harmonic  variations,  we  may 
name  it  harmonic  ecjualiti/  of  variations,  or  of  percentages.  And 
when  we  have  the  equality  which  appears  in  geometric  varia- 

^  For  the  universal  demonstration  see  Appendix  B,  II. 

*  These  fonniihe  make  it  clear  that  the  al)Ove-used  combinations  exhaust  all 
the  possible  ways  of  reckoning  percentages  and  of  getting  equality  between  varia- 
tions. 


'2'-]x     (iKNKh'A  I.   .\i;(MMi;.\r   i-oi;    riii':  ckomktkic   mkan 

ti<tii>,  WT  in;i\  name  it  (/(■(iiiicfric  efjiia/ifi/  ol"  variations,  or  of 
percentages. 

Of  tlie  two  t>eneral  kinds  of  equality  wliieh  may  run  throug'li 
all  these  varieties  of  variations,  the  ecj^nality  of  proj)ortion  must 
ahvavs  be  present  if  the  variations  are  to  be  as  we  have  been 
describmg  them  [i.  e.,  if  their  ])crcentages  are  to  be  equal  in  the 
three  ways  described).  But  tlu'  equality  of  diiference,  or  of  dis- 
tance traversed,  need  l)e  })resent  only  under  certain  conditions, 
dirt'erent  in  each  of  the  three  varieties.  Of  course  when  we 
have  two  \ariations  presented  merely  as  variations,  we  may 
twist  abont  the  starting  and  ending  points  as  we  please,  and  so 
whenever  we  ha\e  an  ecjuality  of  proportion,  we  may  also  con- 
ceive of  a  corresponding  ecpiality  of  difference.  If,  however,  the 
variations  are  given  us  as  the  detinite  variations  of  an  attribute 
in  certain  things,  which  have  started  from  certain  figures  and 
have  ended  at  certain  figures,  then  an  equality  of  pro])ortion 
may  be  present,  in  either  of  the  three  forms,  without  the  corre- 
sponding equality  of  difference,  although  the  latter  cannot  be 
present  without  the  former.  The  conditions  for  the  presence  of 
the  ('([ualities  of  difference  are  plain.  In  the  case  of  arithmetic 
variations,  the  two  things,  or  individuals,  must  be  equally  large 
(or  importiint)  at  the  first  period,  and  at  this  period  only  ;  for 
thev  both  start  from  unity,  or  fnmi  any  other  connnon  figure, 
that  is,  from  the  same  level  of  e([uality  in  size  or  anionnt.  In 
the  case  of  harm(»nie  variations,  the  two  things,  or  individuals, 
must  be  ecjnally  large  (or  important)  at  the  second  period,  and 
then  onlv  ;  for  they  both  end  by  being  e(jual  to  iniity,  or  to 
some  other  e(»nuHon  fignre,  that  is,  they  come  to  the  same  level 
of  size.  In  the  case  of  geometric  variations,  the  two  things,  or 
individuals,  nuist  be  alternately  e(|ual  in  size,  the  one  at  the  first 
]»eriod  with  the  other  at  the  sec^ond,  and  again  the  former  at  the 
second  with  the  latter  at  the  first,  so  that  they  are  equal  in  size 
over  both  the  periods  to(/ether.  The  connection  between  these  con- 
ditions and  the  subject  of  weighting  is  obvious. 

When  we  have  these  e(|nalities  not  merely  of  proportion  but 
of  distances  aetnally  traversed,  we  virtually  have  three  different 
kinds  of  eonqx'iisation — arithnietir,  harmonic  and  (jeoinetric  com- 


KC^rALI'l  ^'     IN     OIM'OSITK     VA  IMA  TIONS  2.">1> 

jirn.sdflon.  Tliere  may  be  airreoiiicnt  between  these  at  times,  as 
^\(■  shall  sec,  aiid  also  disa<2:reemeiit.  In  tlie  latter  case  it  is  the 
jn'oblem  befoiv  iis  to  decich'  which  of  these  eonipensations  is  the 
jn'oper  one  for  our  subject. 

§  5.  So  far  we  have  been  dealing  with  only  two  things  (or 
cUisses)  that  \ary,  and  all  these  relations  have  been  found  U)  be 
universal  when  we  are  dealing  with  two  things  (or  classes),  with 
even  weighting  in  the  averaging  of  their  variations.  All  of 
them  are,  or  may  be,  universal,  when  unity  is  the  arithmetic, 
harmonic  or  geometric  mcdn  between  two  opposite  variations. 
Furthermore,  now,  they  are  completely  and  unconditionally  uni- 
versal in  the  cases  of  the  arithmetic  and  harmonic  averages,  no 
mutter  how  many  variations  on  the  one  side  be  opposed  to  no 
matter  how  many  on  the  other.  In  these  cases  if  unity  be  the 
aritlmietic  or  the  harmt)nic  average  (in  each  case  with  the  weight- 
ing of  the  period  proper  for  it)  between  all  the  variations,  there 
is  equality  of  proportion  between  all  the  variations  ou  the  one 
side  together  and  all  the  variations  on  the  other  side  together, 
and,  with  observance  of  the  proper  starting  and  finishing  points, 
there  is  e({uality  between  the  distance  traversed  by  all  the  rising- 
things  and  the  distance  traversed  by  all  the  falling  things,  the 
whole  distance  in  each  case  being  obtained  by  summing  up  the 
distances  traversed  by  all  the  things  individually. 

Thus  in  the  case  of  compound  arithmetic  variations,  if  in- 
stead of  one  B  we  have  two  B's,  and  both  fall  together  from  1.00 
in  the  compensation  for  the  rise  of  A  from  1.00  to  l.oO,  the 
figure  to  which  they  fall  must  be  such  that  the  arithmetic 
average  between  it  twice  repeated  and  1.50  is  uiiity.  This 
figure  is  .75  ;  for  |^(1.50  -f-  "2  x  .75)  =  LOO.""'  Or  if  two  things 
fall  une(|ually,  suppose  the  one  has  fallen  to  .80,  then  the  other 
must  fall  to  a  figure  between  which  and  .80  and  1.50  the  arith- 
metic average  is  unity.     This  is  .70  ;  for  J(1.50  -(-  .80  -(-  .70) 

'  It  should  be  noticed  that  the  two  figures,  1.50  and  .75,  are  the  simple  harmonie 
extremes  around  1.00,  but  the  three  figures,  1.50  and  .75  twice  repeated,  are 
arithmetic  extremes  around  1.00.  The  figure  .75  repeated  a  fractional  number 
of  times  may  even  be  a  geometric  extreme  opposed  to  1.50  ;  for  we  may  have 

'  V'^  1.50  ^^75^  =  1 .00,  and  it  is  not  difficult  to  find  j-  At  is  -  ,"-^J  =  1.40S(iy 


24(t     (;i:nki;al  aimmmknt  foi;  tiik  (;i:().Mi:ri!i<    mean 

=  1.00.  .\ii(l  similarly  in  other  oases.''  Here  the  sums  of  the 
percentage  of  the  two  falls,  namely  '2')  +  2."),  and  20  +  -"JO,  are 
equal  to  the  pereeutage  of  the  one  rise,  namely  oO  ;  and  the 
sums  of  the  distances  traversed  by  the  two  falliuu'  thinus,  namely 
,2o  +  .2o,  and  ,20  -f  .30,  are  equal  to  the  distance  traversed  l)y 
the  one  rising  thing,  namely  .50.  These  equalities  exist — the  latter 
if  all  the  variations  start  from  the  same  figure — whatever  be  the 
number  of  the  arithmetically  compensating  variations  on  either 
side.     The  general  f(»rmnla  with  n  etjually  rising  and  n'  equally 

falling  things  is  n  I     " '  j  =  y/  (     ^  .      ?  J  ;    and  if  a^  =  6,, 

For  compound  harmonic  variations,  in  a  similar  supposition, 

the  fonnula   is  /(  (  — ^  |  =  ^i'  (  -^-y — -  \  ;    and   if    a.,  =  />.., 

n  (a.,  —  (i^)  =  »'(^i  —  6.,).  For  example,  the  rise  of  A  alone  from 
1.00  to    1.50  is  harmonically  compensated   by  the  equal    falls 

of  two  B's  from  1 .75  to  1.50  ;  for  ,    ,     -^^ ^t^ttt  =  l-'><*- 

^   ^  \00  1.75^ 

50  +  "  ^  1 .50 
Or  if  one  thing  has  already  fallen  from  say  l.SO  to  1.50,  another 

1 

must    fall    from    1.70    to    l.oO  ;    tor 


1  /l.O 
3A1.5 


1  /l.O 
3  \lJ) 


1.00      1.80      1.70 
50  "'"1750"''  1.50 


=  1.00.  And  so  in  other  cases.''  ITere,  too,  the  sums  of  the 
percentages  of  the  two  falls,  measiu'cd  from  the  ending  point, 
which  are  IG?  _)_  Ifji,  and  20  +  13  J,  are  equal  to  the  percen- 
tage of  the  one  rise  similarly  measin-ed,  which  is  33^  ;  and  the 
sums  of  the  distances  traversed  by  the  two  falling  things, 
namely  .25  -f  .25,  and  .30  +  .20,  are  etpial  to  the  distance  trav- 
ersed by  the  (»ne  rising  thing,  namely  .50.  .Vnd  these  equalities 
exist — the  latter  if  all  the  variations  end  at  the  same  figure — 
whatever  be  the  number  of  the  compensating  variations  on 
either  side. 

®  If  the  one  has  already  fallen  to  .50,  the  other  must  remain  at  l.Od.  If  the  one 
has  already  fallen  beyond  ..")(),  th(!  other  must  rise. 

■^  If  the  one  has  already  fallen  from  2.00  to  l.oO,  the  other  must  remain  at  l..')(i. 
If  the  one  has  fallen  from  above  2.00,  the  other  must  rise  to  1.50  from  below. 


EQITAJJTY     IX    OIM'OSITK    \' A  lU  A'llON.S  241 

^  ().   But  in  the  compound  geometric  variations  the  equality 
of  proportion  ceases  to  exist,  and  the  e([uality  of  distances  trav- 
ersed exists  only  at  the  sacrifice  of  some  of  the  relations  pre- 
viously included,  so  that  we  no  longer  have  the  same  universal 
relations  as  before.     This  becomes  evident  the  moment  we  deal 
with   two   equal  variations  in   compensation   for  one  variation. 
Thus  the  rise  of  A  from  1.00  to  1.50  is  geometrically  compen- 
sated by  the  falls  of  two  B's  from  1.00  to  a  figure  between  which 
twice  repeated  and  l.oO  the  geometric  average  is  unity.     This 
is    .8165   (=l/l;   for  i>/TT5(r>rT81652=  1.00.      Here  the 
percentage  of  the  fall  from  1.00  to  .<S1()5   reckoned  in  1.00  is 
18.35,  and  twice  this  is  more  than  the  percentage  of  the  rise  of  A 
reckoned  in  1 .50,  which  is  33  J  ;  and  the  percentage  of  the  fall 
reckoned  in  .8105  is  22.47,  and  twice  this  is  less  than  the  per- 
centage of  the  rise  reckoned  in  1 .00,  which  is  50.     As  regards 
the   distances   traversed,    the    variations   l)eing   kept   the   same 
merely  as  variations,  the  compensation  may  be  by  the  two  B's 
falling  from  1.2247  to  1.00,  or  from  1.50  to  1.2247.     Then  in 
the  former  case  they  together  traverse  a  shorter,  and  in  the  latter 
a  greater,  distance  than  A.      Now  if  the  one  thing  falls  from 
1.50  to   1.2247  and  the  other  from  1.2247  to  1.00,  they  both 
together  traverse  the  same  distance  as  A,  covering  in  two  stages 
downwards  the  same  road  covered  by  A  in  one  leap  upwards. 
But  then  the  two  B's  are  not  equal  things  belonging  to  the  same 
class  (they  are   rather  B  and  C,  and  unequal  in  size  over  both 
the  periods  together  so  that  we  should  not  be  justified  in  geo- 
metrically averaging  the  three  variations  with  even  weighting). 
It  is,  however,  possible  for  the  two  B's  to  be  alike,  and  falling 
together  in  the  same  proportion  as  before,  to  traverse  each  half 
the  distance  traversed  by  A.     This  is  when  they  both  fall  from 
1.3623  to  1.1123.     But  then  the  two  B's  are  not  each  equally 
large  or  important  with  the  one  A  over  both  the  periods  to- 
gether ;    for   the  geometric   mean   between   their   starting   and 
ending  points  is   1.23096,  while  the  geometric  mean   between 
the  starting  and  ending  points  of  A  is  1.2247.     Thus  the  rela- 
tions that  hold  true  of  simple  geometric  variations — variations 
of  two  equally  important  things  (or  classes),  with  even  weight- 
16 


242     GENERAL    ARGUMENT    FOR    THE    GEOMETRIC    ilEAN 

iug,  SO  that  unity  is  the  geometric  viean  between  them,  do  not 
hold  true  of  compound  geometric  variations — variations  between 
several  opposed  things,  with  various  weigliting,  such  that  unity 
is  a  geometric  average  between  them.  Wliat  has  above  been 
called  geometric  equality  of  variations  or  of  percentages  existsf 
only  between  opposite  variations  of  two  things  equally  large  or 
important  over  both  the  periods,  and  does  not  exist  in  more  than 
two  variations  that  geometrically  compensate  for  one  another 
(unless  they  do  so  in  pairs).  That  this  is  so  in  general  may  be 
seen  from  the  formulse.  AVith  several  things,  /;  in  number, 
falling  together  in  compensation    for  the  rise  of  A  alone,  the 

formulae  are  either-^ =    ^^      '\  which  is  true  only  if  a^ 

^  62  =  1,    whereupon    a,^  —  o^  =  b^"  —  6,„    and    a.,  =  h^  ;    or 

a  —  a        h  —f).;      ,  .  ,    .    ^  ,     .,,  ,         ,       , 

_f^ i  __    '        -  ^  which  IS  true  only  11  a.,  =  ^^  =  1,  whereupon 

a^  0| 

a.^  —  a^  =  6,  —  6.,",  and  a^  =  h.^".  Now  while  A  traverses  the 
distance  a.-^  —  «j,  none  of  the  opposing  things  traverse  the  dis- 
tance b"  —  b.^,  or  6j  —  b^',  but  only  the  distance  6^  —  b.„  which 

is  not  -  the  distance  «„  —  «,.     Still,  trial  shows  that  in  ordinary 

n  21;  J 

cases,  that  is,  with  not  more  than  two  or  three  things  opposing 
any  one  thing,  the  relations  between  compound  geometric  vari- 
ations are  very  nearly  the  same  as  those  above  described. 

What  is  here  discovered,  in  differentiating  the  geometric 
average  from  the  geometric  mean,  and  segregating  it  from  the 
arithmetic  and  harmonic  averages,  which  'behave  universally  in 
the  same  manner  as  do  the  arithmetic  and  harmonic  means,  will 
later  call  for  our  rejMjated  attention.  It  shows  that  we  cannot 
treat  geometric  averages  as  we  do  geometric  means,  and  that  the 
use  of  only  two  figures,  with  even  weighting,  will  lead  us  into  error 
if  we  treat  them  as  examples  good  for  all  cases.  After  examin- 
ing what  is  true  of  the  geometric  mean,  with  two  things  evenly 
weighted,  we  must  always  tui-n  to  more  cjomplex  exam})les ;  and 
in  them  we  sliall  find  that  convenient  relations  discovered  in  the 
former  case  no  longer  iiold — except,  in  ordinary  cases,  only  ap- 
])roximately.  \\  e  thus  have  obtiuned  a  warning  for  all  our 
future  labors. 


EQI'ALITV    IN    Ol'l'O.SITE    VAUIATIO.NS  243 

It  is  pleasing  to  iKitice  that  in  the  employment  of  the  geo- 
metric mean  already  made  in  Chapter  IV.  we  have  nothing  to 
revise ;  for  we  there  dealt  with  the  geometric  mean  proper, 
that  is,  with  two  figures — two  periods — regarded  as  equally 
important. 

And  we  may  continue  for  the  present  to  treat  principally  of 
means. 

§  7.  Now  in  our  own  special  subject  people  have  generally 
confined  their  attention  to  price  variations,  and  these  price  varia- 
tions they  have,  of  course,  measured  in  the  usual  way,  conceiv- 
ing of  them  all  as  starting  from  the  same  figure.  They  have 
also  generally  used  the  arithmetic  average,  assuming  that  the 
proper  equality  in  com]>ensatory  changes  is  the  arithmetic.  The 
adoption  of  the  arithmetic  average  we  have  seen  to  be  due  prin- 
cipally to  its  convenience.  But  when  people  have  looked  for  a 
reason  to  justify  its  use,  they  have  lighted  upon  the  fact  that  by 
it  alone — measured  in  the  usual  way — does  equality,  especially 
the  equality  of  distance  traversed,  exist  in  the  opposite  varia- 
tions that  are  supposed  to  compensate  for  each  other.  We  now 
see  that  it  is  only  because  of  the  method  usually  adopted  for 
reckoning  percentages  that  equality  is  apparent  only  in  the  arith- 
metic compensations ;  and  yet  this  is  only  one  of  three  possible 
methods.  The  usual  method  is  the  most  convenient  and  the 
most  suitable  for  measuring  the  proportions  of  the  variations ; 
but  it  is  no  better  than  either  of  the  others  for  measuring  the 
actual  amounts,  or  distances  traversed,  of  the  variations.  We 
therefore  find  ourselves  in  need  of  special  reasons  in  our  subject 
for  showing  why  the  one  of  these  sets  of  starting  points  is  the 
proper  one  rather  than  another.  But  for  this  purpose  we  must 
appeal  to  facts  ;  for  only  facts  will  show  what  the  starting  points 
really  are.  Facts  will  give  us  the  weighting,  and  the  ])eri()d,  or 
periods,  of  the  weighting.  And  the  mean,  or  average,  will  then 
have  to  be  chosen  with  dependence  upon  the  period,  or  periods, 
whose  weighting  is  chosen.  Otherwise  our  procedure  will  be 
arbitrary. 

In  general  these  principles,  derived  from  what  has  above  been 
shown  at  the  end  of  §  4,  may  be  stated  : — (1)  If  it  happens  that 


244    (;i:m;i;.\[.  AiaiiMKxr  for  the  (iKoMETRic  >rKAN 

the  thiiiiis  that  vary  arc  c(|ual  at  th(^  start  and  diverge,  using; 
oven  weiti'htinu',  we  sliould  use  the  aritlimetie  mean.  ("2)  If  it 
haj)i)ens  that  the  things  are  une(|ual  at  the  start  and  converging 
become  equal  at  the  finish,  using  even  weighting,  we  should  use 
the  harmonic  mean.  (-">)  If  it  haj)i)ens  that  the  things  are  un- 
ci [ual  at  the  start  and  at  the  end  are  reversely  unequal,  having 
exactly  changed  ]>laces,  then,  using  even  weighting,  we  should . 
use  the  geometric  mean.  Or,  putting  the  matter  still  more 
broadly,  whatever  the  weighting  be,  provided  we  are  dealing 
with  the  same  material  things  at  both  periods,  (1)  if  we  use  the 
weighting  of  the  first  period  we  should  use  the  arithmetic  aver- 
age ;  (2)  if  we  nse  the  weighting  of  the  second  period,  we  should 
use  the  harni()ni{;  average ;  (3)  if  we  use  the  weighting  of  both 
periods,  we  should  use  the  geometric  average,  although  in  this 
case  only  approximation  to  a  correct  result  may  be  expected.  A 
peculiar  relation  exists  between  these  three  means  (or  averages) 
so  used  (applied  to  the  same  things  at  both  periods),  which  it  is 
not  the  place  here  to  jjoint  out,  but  which  will  be  shown  in  a 
later  Chapter.  Some  of  these  principles  are  applicable  some- 
times also  to  classes  that  are  not  composed  at  both  the  periods  of 
the  same  things.  The  examination  of  these  cases,  so  important 
in  our  subject,  must  likewise  be  postponed. 

Now  if  we  simply  suppose  we  are  dealing  with  equally  large^ 
or  equally  im|)()rtant,  things,  or  classes,  the  natural  ini])licatiou 
is  that  tiiey  are  equally  large  or  inq)()i-tant,  not  at  only  one  of 
the  periods,  but  eitiier  at  both,  or  somehow  over  both  together. 
Therefore  in  sucsh  general  suppositional  cases  the  proper  mean  or 
average  to  use  with  even  weighting  is  the  geometric. 

§  8.  Furthermore,  in  our  subject  we  should  remember  that 
we  have  to  look  at  the  variations  in  two  asjiects.  Now  if  a 
person  finds  to  his  own  satisfaction  that  the  arithmetic  average, 
with  a  certain  wei'>;htino:,  is  the  ri(2:iit  one  for  the  averau'ino:  of 
price  variations,  because,  perhaps,  of  the  existence  of  equality  of 
distance  traversed  in  these  variations  Avhen  supposed  to  be  com- 
pensatoiy,  he  will  have  to  use,  with  the  same  weighting,  the 
liarmonic  average  for  the  variations  of  the  exchange-values  of 
money,  or  of  the  mass-([uantities  purchased,  in  spite  of  the  fact 


NATURES    OF    THE    SUBJECTS  245 

that  the  equality  of  distance  traversed  is  now  lost  (although,  in 
certain  cases,  as  we  shall  see,  he  may  still  use  the  arithmetic 
average,  but  with  difFerent  weighting).  And  reversely,  if  he 
prefers  the  arithmetic  average  for  averaging  the  variations  of 
"the  exchange-values  of  money,  he  will  have  to  use,  with  the 
same  weighting,  the  harmonic  average  in  averaging  the  price 
variations,  in  sj)ite  of  the  loss  here  of  equality  of  distance  trav- 
ersed (although,  in  certain  cases,  as  before,  he  may  still  use  the 
arithmetic  average,  but  with  different  weighting).  Hence  each 
of  these  persons  ought  to  be  called  upon  to  give  other  reason 
for  his  position  than  the  mere  ])reference  for  arithmetic  equality, 
since  he  can  have  this  (except  in  certain  cases)  only  in  one  of 
the  two  aspects  of  the  problem.  He  should  state  also  why  the 
harmonic  average  is  to  be  preferred  on  the  other  side  of  the 
problem.  It  is  only  the  advocate  of  the  geometric  average  who 
can  use  the  same  average,  with  the  same  Mcighting,  and  the 
same  arguments  for  it,  on  both  sides  of  the  problem.  And  if 
his  arguments  should  suffer  by  comparison  with  the  arguments 
for  the  arithmetic  average,  they  might  gain  by  comparison  with 
the  arguments,  or  lack  of  arguments,  for  the  harmonic  average, 
which  the  others  are  also  called  upon  to  advance.  As  no  argu- 
ments have  ever  been  adduced  for  the  harmonic  average  directly, 
we  shall  generally  have  to  compare  the  arguments  for  the  geo- 
metric average  only  with  the  arguments  for  the  arithmetic 
average. 

II. 

§  1.  We  shall  now  turn  our  attention  to  the  nature  of  the 
subjects  to  which  the  different  averages  may  be  ap})licable. 

In  some  subjects  the  zero  point  is  a  purely  arbitrary  position, 
or  if  not  arbitrary,  it  is  at  least  a  point  that  can  be  passed. 
When  a  positive  figure  passes  such  a  zero,  it  is  to  be  treated  as 
negative  ;  and  a  negative  figure,  passing  it,  becomes  positive. 
A  familiar  example  is  the  figure  representing  tenq)erature  in 
our  usual  methods  of  measuring  temperature,  which  figure  may 
be  above  or  below  zero.  In  all  these  cases  there  is  no  reason 
apparent  why  compensation  l)y  arithmetically  equal   variations 


246      GHXERAL    ARttUMKXT    FOR    TIIK    GEOMETRIC    MEAN 

should  not  be  the  pntper  one  ;  and  in  fact  wc  generally  find  it 
to  be  so.  The  following  is  an  example.  On  a  inatheraatical 
balance  two  equal  weights  are  in  equilibrium  at  equal  distances 
on  eacti  side  of  the  fulcrum,  which  point  is  represented  by  zero, 
the  one  weight  being  at  the  distance,  say  +  a,  from  it  on  the 
ritrht,  and  the  other  at  the  distance  —  a  from  it  on  the  left. 
Now  if  the  weight  on  the  right  be  split  into  two  equal  })arts 
and  the  one  of  these  be  moved  further  to  the  right,  the  other, 
in  order  to  keep  the  e(|uilibrium,  must  be  moved  to  the  left  to- 
ward the  fulcrum  at  an  equal  speed,  the  two  always  being  at 
equal  distances  from  their  original  position,  whose  distance  from 
the  fulcrum,  +  a,  is  thus  the  arithmetic  mean  between  their  dis- 
tances from  the  fulcrum.  This  is  possible  because  when  the 
part  moving  to  tlie  right  passes  the  distance  +  2  a,  the  other  part 
moving  to  the  left  may  pass  the  zero  point.  Thus,  for  instance, 
when  the  former  is  at  -f-  3  a,  the  latter  will  be  at  —  a,  and  these 
are  the  arithmetic  extremes  around  -|-  (/.' 

Again  there  are  subjects  in  which,  while  the  one  thing,  in 
falling  in  some  attribute,  cannot  pass  the  zero  point,  the  other 
thing  in  rising  likewise  can  not  pass  a  definite  point  equally  far 
above  the  starting  point  (supposing  even  weighting  to  be  proper). 
For  instance,  if  two  parts  of  the  territory  of  a  country  are 
e({ually  populous  and  the  population  of  the  one  part  gradually 
moves  over  to  the  other  part,  as  the  population  of  the  former 
decreases  toward  zero,  the  |)opulation  of  the  latter  rises  toward 
double, — the  total  being  supposed  to  remain  the  same  ; — and  so, 
the  popidation  of  each  half  being  represented  as  1  at  the  start, 
as  the  one  cannot  sink  below  1  beyond  0,  the  other  cannot  rise 
above  1  beyond  2.  In  this  movement,  the  po})ulation  of  the 
whole  country  remaining  the  same,  the  average  population  of  its 
parts  remains  the  same.      This  constancy  is  observed  by  drawing 

^  Here  we  are  using  even  weighting  l)eeause  the  two  parts  arc  equal.  If  the 
one  were  half  the  otlier,  /.  e.,  if  the  original  weigiit  were  divided  into  three  equal 
parts  and  two  of  tiiem  move  together,  the  latter  would  have  to  move  only  lialf  as 
fast  as  the  former.  For  instance,  +  a  is  the  aritlimetic  mean  between  (-  3  a  twice 
repeated  and— ;}«;  and  when  the  larger  part  has  moved  from  +  a  to  +  3  a, 
traversing  the  distance  +3a— a  =  2«  (wherefore  each  of  its  halves  traversing 
tills  distance,  the  two  halves  together  have  moved  over  i-4a),  the  smaller  part 
(equal  to  each  of  the  halves  of  the  other)  has  passed  from  +  a  to  —  3 a,  travers- 
ing tiic  distance      '.i  a  —  a  =  —  4 a. 


NATURES    OF    THE    SUBJECTS  247 

the  arithmetic  average  of  the  variations,  with  even  weigliting.^ 
And  the  compensation  is  by  arithmetic  equality,  with  equality 
of  distance  traversed  away  from  the  common  starting  point. 

Reversely  there  are  subjects  in  which  a  common  point  may  be 
approached,  on  opposite  sides,  from  points  within  two  impassable 
limits,  the  one  of  these  being  zero  and  the  other  some  definite 
figure.  Thus  in  the  preceding  example  reversed,  if  the  popula- 
tions in  the  two  parts  are  unequal  at  first  and  by  the  moving  of 
people  from  the  larger  to  the  smaller  become  equal,  then,  the 
total  population  remaining  unchanged,  the  constancy  of  the 
average  of  the  two  j)arts  will  be  found  to  be  indicated  by  the 
harmonic  average,  with  even  weighting.^  Here  the  compensa- 
tion is  by  harmonic  equality,  or  equality  of  distance  traversed 
to  the  common  ending  points. 

But  there  are  subjects  in  which  the  zero  point  is  absolute  and 
impassable,  a  negative  quantity  having  no  meaning,  or  there 
being  nothing  that  can  be  conceived  of  as  negative,  while  on  the 
rising  side  there  is  no  fixed  limit.  In  these  the  use  of  the  arith- 
metic average  for  measuring  compensatory  variations  is  obviously 
out  of  place,  because  the  conditions  here  permit  the  existence  of 
variations  on  the  one  side  which  cannot  be  arithmetically  com- 
pensated on  the  other.  It  is  evident  that  a  variation  moving  in 
an  unlimited  direction  is  not  properly  compensated  by  a  varia- 
tion moving  at  the  same  speed  toward  a  fixed  limit.  The  arith- 
metic average  is  admissible  only  in  the  two  cases  just  reviewed  : 
— either  where  there  are  no  limits  on  either  side,  or  where  there 
are  fixed  limits  on  both  sides.  Then  the  two  limits  should  be 
reached  at  the  same  moment  by  tlie  outward  moving  quantities. 
Now  when  zero  is  a  limit  on  the  one  side  and  on  the  other  there 
is  no  finite  limit,  we  can  conceive  of  infinity  as  being  the  limit 
on  this  side.     Compensatory  variations  would  then  be  such  that 

2  Even  weighting  is  to  be  employed  because  of  the  equality  of  the  population 
in  the  two  parts  at  the  first  period.  If  the  one  part  were  larger  than  the  other  at 
the  first  period,  its  weight  would  have  to  be  proportionally  larger. 

3  Even  weighting  is  employed  because  the  populations  of  the  parts  are  equal  at 
the  second  period.  If  they  finish  with  dift'erent  sizes,  uneven  weigliting  accord- 
ing to  these  must  be  employed. — The  mathematical  reader  will  perceive  an 
agreement,  in  such  cases,  between  the  the  two  different  avei-ages  with  the  two 
different  weightings.  This  is  part  of  the  peculiarity  above  alluded  to,  M'hich  will 
be  examined  later  (in  Chapter  X.). 


248      GENERAL    ARGT'MENT    FOR    THE    GEOMETRIC    MEAN 

the  one  approaches  infinity  at  the  same  rate  as  the  other  ap- 
proaches zero.  Such  are  geometric  variations.  In  these  the 
falling  (piantity  reaches  zero  no  sooner  than  the  other  reaches 
infinity.  Hence  whenever  we  have  these  limits,  the  geometric 
average  is  tlie  proper  one  for  measnring  compensatory  varia- 
tions, or  for  indicating  their  combined  results. 

§  2.  Now  exchange- values,  and  }>rices,  are  exactly  such  (pian- 
tities.  They  can  rise  infinitely.  I  hey  cannot  sink  below  zero 
or  nothing.  It  is  true  that  we  sometimes  pay  to  get  rid  of  a 
thing,  as  a  disease,  or  an  obstruction  of  any  sort.  Such  things 
may  be  represented  as  having  negative  exchange-value,  or  nega- 
tive price.  Also  things  that  have  positive  exchange-value  may 
sink  to  negative  exchange-value  either  through  deterioration  or 
through  superfluity.  But  there  are  few  instances  of  things  that 
have  positive  exchange-value  sinking  to  negative  exchange-value 
when  new.  And  even  if  they  were  conunon,  nobody  would  claim 
that,  [A]  and  [B]  l)eing  two  ecjually  useful  classes  of  things  diffi- 
cult of  attainment,  if  [A]  should  more  than  double  in  price,  this 
could  be  compensated  by  [B]  becoming  a  repugnant  object  difficult 
of  avoidance.  Instead  of  remaining  in  a  constant  position  with 
respect  to  these  two  objects  together,  we  should  be  doubly  in- 
commoded, as  we  should  have  to  pay  more  to  procure  the  one  and 
additionally  j)ay  to  get  rid  of  the  other.  And  when  a  class  of 
things,  through  a  change  in  our  wants,  falls  to  zero  in  price  or 
exchange-value,  it  is  no  longer  produced,  it  becomes  an  extinct 
class,  and  is  no  more  to  be  taken  into  account  in  our  comparisons. 

Therefore  no  rise  of  A  in  price  from  1 .00  to  a  height  above 
2.00  can  be  arithmetically  compensated. by  any  fall  of  B  from 
1.00,  if  they  belong  to  equally  im])ortant  classes,  since  B  would 
have  to  fall  below  zero,  which  here  is  meaningless.  Hence  in 
these  cases  there  can  be  no  arithmetic  comjiensation. 

As  regards  the  harmonic  compensation,  the  trouble  is  inverted. 
We  have  seen  that  in  this  kind  of  compensation,  if  B  falls  in 
price  from  1 .50  to  1 .00,  A  should  rise  in  price  from  .50  to  1 .00. 
Then  if  B  fidls  in  price  from  2.00  to  1.00,  A  should  rise  in 
price  from  0  to  1.00.  And  if  B  falls  in  price  from  any  figure 
above  2.00  to   1.00,  that   is,  if  it  falls  by  more  than   half,  A 


APPLIED    TO    EXCHANGE-VALUES  249 

should  rise  in  price  from  below  zero  to  1.00,  whicli  is  impos- 
sible. Hence  in  such  cases  there  can  be  no  harmonic 
compensation. 

Similarly  from  the  jioint  of  view  from  which  our  problem 
must  be  regarded,  if  the  exchange-value  of  money  in  [B]  rises 
by  more  than  100  per  cent.,  the  compensatory  fall  of  the  ex- 
change-value of  money  in  [A]  should  be  to  below  zero  accord- 
ing to  the  arithmetic  method  of  avcrao-ino;,  which  therefore  is  in- 
applicable  in  this  case.  And  if  the  exchange-value  of  money  in 
[A]  falls  to  less  than  half,  the  exchange-value  of  money  in  [B] 
should  rise  from  below  zero  according  to  the  harmonic  method 
of  averaging,  which  therefore  is  inapplicable  here. 

But  in  the  use  of  the  geometric  compensation  there  are  no 
such  impossible  cases.  If  the  ])rice  of  A  rises  from  any  figure 
to  any  figure,  we  merely  have  to  suppose  that  the  price  of  B  falls 
from  the  latter  to  the  former  figure,  or  in  the  same  })roportion. 
And  if  the  price  of  B  falls  from  any  figure  to  any  figure,  we 
merely  have  to  suppose  that  the  price  of  A  rises  from  the  latter 
to  the  former  figure,  or  in  the  same  proportion.  Similarly, 
though  inversely,  in  the  case  of  the  exchange-values  of  money 
in  the  two  classes.  There  is  no  conceivable  case  in  which  this 
kind  of  compensation  is  not  possible — that  is,  on  the  supposition 
we  are  all  along  making,  in  our  problem  about  exchange-values 
or  prices,  that  the  classes  are  equally  large  or  important  at,  or 
over,  both  the  periods  together. 

III. 

§  1.  Having  examined  the  general  nature  of  the  three  means 
and  averages  and  of  the  subjects  to  which  they  are  applicable, 
we  may  now  examine  the  means  and  averages  as  applicable  to 
our  special  subject. 

The  advocate  of  the  arithmetic  averaging  of  prices  says  that 
when  the  class  [A]  rises  in  price  by  a  certain  percentage,  in 
order  to  compensate  for  this  rise  and  to  keep  the  average  of 
prices  constant,  the  class  [B]  should  fiill  by  an  equal  percentage. 
Usually  the  writers  on  this  subject  have  not  much  concerned 
themselves  about  weighting,  but  when  their  attention  is  turned 


250      (JENERAI.    ARGUMENT    FOR    THE    GEOMETRIC    MEAX 

toward  it,  tliey  posit  merely  tliat  the  classes  in  (juestion  should  be 
equally  important.  They  have  not  entered  into  the  question  at 
what  period  the  classes  should  be  equally  im])ortant.  AVe  may, 
then,  at  present,  take  our  own  position  on  this  subject,  and  re- 
quire that  the  classes  be  ccpially  im])ortant,  or  large,  over  both 
the  periods  together — either  alternately,  or  constantly  so,  or  on 
the  average. 

AW'  may  suppose  that  as  A  rises  in  price  by  one  per  cent,  at 
a  time  (always  reckoned  on  the  original  starting  point  at  1.00), 
B  foils  by  one  per  cent,  at  a  time  (likewise  always  reckoned  on 
the  same  original  starting  point  at  1.00).  According  to  the 
arithmetic  averagist  these  opposite  movements  always  leave  the 
exchange-value  of  money  and  the  level  of  prices  unchanged — 
until  2.00  is  reached  by  A,  after  which  he  does  not  concern 
himself  further  (or  requires  that  another  equally  important  class 
shall  begin  to  fall).  But,  now,  every  successive  rise  of  A  is  a 
smaller  rise  reckoned  from  /fe  starting  point,  while  every  suc- 
cessive fall  of  B,  similarly  reckoned,  is  a  larger  fell.  Thus, 
while  the  first  rise  and  fall  are  each  by  1  per  cent.,  the  second 

/1.02  -  1.01\ 
rise  of  A  from    1.01   to   1.02   is  a  rise  by  100  (  --^-j \ 

=  0.99001*1)  ])er  cent.,  and  the  corresponding  fall  of  B  from  .99 

(99 98\ 
^^^^  j=  1 .0101  per  cent.     The  dif- 
ference will  be  shown  more  plainly  by  an  extreme  case.    When 
A  has  risen  to  1.98  and  B  fallen  to  .02,  the  next  rise  of  A  to 

.      ,             /1.99-1.98\      ^  ^^_  ,        , 

1.99  IS  a  rise  by  100  (        j^ 1=  0.50o  per  cent. ;  but  the 

/.02-.01\ 
next  fall  of  B  to  .01  is  a  fell  by  100  (        ^^       j=  50  per  cent. 

Now  suppose  we  begin  with  things  in  this  condition.  The  i)rice 
of  A  being  1.98,  the  price  of  0.505  A  is  1.00,  and  the  price  of  B 
being  .02,  the  price  of  50  B  is  1.00.  We  may,  then,  form  a  new 
unit  for  each  of  these  quantities,  say  A'  priced  at  1.00  and  B' 
[)ri('ed  at  1.00 — and  we  may  still  supjiose  that  the  classes  [A] 
and  [B]  are  cc^ually  important.  Then  the  arithmetic  averagist, 
if  he  says  the  rise  of  the  price  of  A  to  1.99  is  compensated  by  a 


APPLIED    TO    EXCHANGP>VALUE.S  251 

fall  of"  the  price  of  B  to  .01,  must  .say  that  the  rise  of  the  price 
of  A'  to  1.00505  is  compensatecl  by  the  fall  of  the  price  of  B' 
to  .50  ;  which  is  absurd  ou  his  own  principles.  Thus  the  posi- 
tion of  the  arithmetic  averagist  leads  him  into  inconsistency. 
And  on  ffoino^  back  to  the  start,  we  now  see  that  even  the  first 
fall  from  1.00  to  .99  is  a  larger  fall  than  the  rise  from  1.00  to 
1.01,  if  we  analyze  each  of  these  variations  into  component 
stages.  Therefore  all  such  price  variations  mean  an  apprecia- 
tion of  money. 

Ou  the  other  hand,  in  harmonic  variations  the  fall  demanded 
in  compensation  for  a  rise  is  always  smaller  than  the  rise.  This 
is  so  obvious  that  it  does  not  need  explication.  But  the  position 
of  the  harmonic  averagist  of  prices  may  be  shown  to  be  wrong 
also  in  this  way.  His  position  is  that  we  should  have  arithmetic 
equality  in  the  compensatory  changes  in  the  quantities  of  things 
purchasable,  whereby  the  exchange-value  of  money  in  their 
classes  is  measured.  Here  of  course  the  same  absurdity  comes 
to  light,  with  the  consequent  inconsistency,  as  in  the  case  where 
the  arithmetic  average  is  applied  to  prices.  And  the  falls  in 
the  exchange- values  of  money  being  too  large,  this  kind  of  varia- 
tions means  a  depreciation  of  money. 

§  2.  The  geometric  method  of  averaging  escapes  such  absurd- 
ity and  such  inconsistency.  It  provides,  not  equality  of  rate  of 
variation  (measured  in  the  usual  way),  but  correspondence.  As 
the  rate  of  the  rise  of  A,  supposed  to  be  always  at  the  same  speed 
from  the  original  starting  ])oint,  grows  smaller  with  every  ad- 
vance, so  the  rate  of  the  fall  of  B  is  made  to  grow  pro})ortion- 
ally  smaller.     Thns  when  A  rises  from  1.00  to  1.01   by  1  per 

cent.,  the  compensatory  fall  of  B  is  from  1 .00  to  =  .990099, 

which  is  a  fall  by  100(1  -  .990099)  =  0.990099  per  cent. 
Then  when  A  next  rises  to  1 .02,  the  compensatory  fall  of  B  is  to 

-— -r  =  ,980392.     Here  the  percentage  of  the  rise  of  A,  reck- 

1    •      •              •            .       ,             1                  / 1.02 -1.01  \ 
oned   in    its  starting  point,  has  sunk  to   100  1 ^--.  |  = 

0.990099,  and  the  percentage  of  the  fall  of   B,   reckoned   in 


252      GEXEUAL    AEGUMKNT    Foi;     I  UK    C  i:<  )M  KTRTC    MKAX 

/.990()!»!)  —  .i)S();5i)2\ 
its   startino-    jxtint,   litis    sunk    to    100  (  (itioo'iq  )^ 

0.9SO407.      Again  when  A  next  rises  to  1.0."),  the  compensatory 

fall  of  B  is  to  =  JITOST.').      Here    the    pereentaoe   of    the 

rise    of    A,    reckoned     in     its     starting     point,    has     sunk     to 

100  (    ■   \  tJtt^^  )  =  0.98O;i!)2  (the  same  as  the  point  to  whicii 

B  previously  fell),  and  the  percentage  of  the  fall  of  B,  reckoned 
in  the  same  way,  has  sunk  to  0.970938  per  cent.  And  so  the 
process  Avill  continue  indefinitely,  the  percentage  of  the  fall  of 
A  always  being  the  same  with  the  point  to  which  B  previously 
fell,  and  the  percentage  of  the  fall  of  B  always  being  still  smaller 
(though  above  the  point  to  which  B  falls).  For  instance,  in  th(! 
ninety-ninth  stage,  the  rise  of  A  from  1.98  to  1.99  is  compensated 

by  the  foil   of  B  from      ^^^  =  .oOoOoO  to  ^  ^^^-  =  .502512,  so 

/I  99  _  1  t)8\ 
that  the  rise  by  100  ( ^     j  =  0.505050  percent,  is  com- 

/.505050-. 50251 2\       ^  _    ^_. 
pensated  by  a  foil   by   100  (  ^  -^ j  =  O.o02o24 

per  cent. 

Now  the  first  comju'iisatory  fall  of  B  from  1.00  to  .990099  is 
the  same  variation  as  a  fall  from  1.01  to  1.00, — merely  the  re- 
verse of  the  rise  of  A  for  which  it  is  offered  in  compensation. 
And  the  next  fiill  of  B  from  .990099  to  .98();i92  is  the  same 
variation  as  a  fall  from  102  to  101,- — again  merely  the  reverse 
of  the  rise  of  A  for  which  it  is  offered  in  compensation.  And 
so  with  all  the  comjKMisatorv  variations  recpiired  by  the  ge(mietric 
averaging  (with  even  weighting).  They  are  merely  the  reverse 
of  each  other.  Whenever  A  rises  from  any  figure,  a^,  to  any 
figure,  «2)  by  a  percentage  (reckoned  in  a^)  obtained  by  dividing 
a  hundred  times  the  difference  >vhich   measures  the  amount  of 

the  rise  l>v  the  earlier  figure,  namelv  100  (    -        '),   the  coni- 

pensatory  i'all  of  B  is  a  fill  from  a.,  to  a^  by  a  ])ercentage  (I'eck- 
oned  in  a„)  obtained  bv  dividino-  a  hundred  times  the  differemre 


APPLIED    TO    EXCHANGE-VALUES  253 

which  measures  the  amount  of  tlie  fall  bv  the  earlier  fimire, 
namely    100  I    "        M.     These  percentages  are  not  the  same, 

but  it  is  evident  that  this  foil  is  equal  to  that  rise.  If  A  rises 
from  ftj  to  a^  ''^^^  then  falls  back  to  a^,  it  is  evident  that  the  fall 
is  equal  to  the  rise,  since  it  brings  A  back  to  its  original  posi- 
tion. Then  the  fall  of  B  from  «.,  to  a^,  equalling  the  fall  of 
A  from  «2  to  a^,  equals  the  rise  of  A  from  a^  to  a,-^  The 
usual  method  of  reckoning  percentages  does  not  manifest  this 
equality,  which  is  shown  mathematically  only  by  reckoning 
the  percentages  from  the  opposite  extremes,  but  in  the  same 
direction.  That  the  mathematical  equality  should  reside  only 
in  this  kind  of  percentage  is  plain,  since  the  compensatory 
variations  must  plainly  be  merely  the  reverse  of  each  other. 
Therefore  the  correctness  of  the  geometric  mean  (although  not 
of  the  geometric  average)  is  demonstrative. 

§  3.  Jevons  wrote  in  his  Prinoiples  of  Seienee : " — "  In  almost 
all  the  calculations  of  statistics  and  commerce  the  o-eometric 
mean  ought,  strictly  speaking,  to  be  used.  If  a  commodity  rises 
in  price  100  per  cent,  and  another  remains  unaltered,  the  mean 
rise  of  a  price  is  not  50  per  cent,  because  the  ratio  150  :  200  is 
not  the  same  as  100  :  150.  The  mean  ratio  is  as  unity  to 
l/l.OO  X  2.00  or  1  to  1.41."  There  is  exaggeration  in  the  first 
part  of  this  statement,  since  the  geometric  mean  is  almost  ex- 
clusively to  be  confined  to  the  measurement  of  variations,  and 
many  calculations  of  statistics  are  not  measurements  of  variations 
, — though  often  providing  the  data  for  such  measurements.  In 
this  statement  we  have  our  problem  solved  in  the  first  form  in 
which  we  approached  it.  Therefore  we  may  find  interest  in 
noticing  this  also.  As  Jevons's  brief  statement  has  not  met  with 
acceptance  on  the  part  of  statisticians  of  prices,  it  needs  to  be 
explicated. 

The  arithmetic  averagist  would  say  that  the  mean  rise  in  the 
above  suppositional  case  is  to  1.50.  Now  suppose,  the  price  of 
B  remaining  at  1.00,  the  price  of  A  rises  first  of  all  to  1.50. 

'  Cf.  the  reasoning  for  Proposition  XI. 
^  Second  ed.,  p.  SOL 


254      GEXEnAI,    ARGUMENT    FOR    THE    GEOMETKMC    MEAN 

Then  the  arithniL'tic  avoi'a«iist  would  say  the  mean  ])i'i('o  has 
risen  to  1.25.  Suppose  that  later  the  priee  of  A  rises  from  1.50 
to  2.()().  This  rise,  measured  from  its  starting  point,  is  a  rise 
by  33  J  per  cent.  Then,  according  to  the  jn'inciples  of  the  arith- 
metic averagist,  the  mean,  already  risen  to  1.25,  ought  to  rise 
further  by  half  of  3„3J,  or  16f,  per  cent,  above  1.25.     This  is 

a  rise  to  1.25  x  l.l'>i  =  1.45433 Therefore  on  his  own 

principles  the  arithmetic  averagist  is  mistaken  in  saying  that 
when  A  rises  to  2.00  the  mean  rise  is  to  1.50.  And  conse- 
quently, too,  he  was  mistaken  when  he  said  it  rose  first  to  1 .25, 
and  his  whole  position  from  beginning  to  end  is  inconsistent 
with  itself,  and  wrong.  The  mean  price  at  every  rise  is  below 
the  arithmetic  mean. 

On  the  other  hand,  the  mean  price  is  always  above  the  liar- 
monic  mean,  because  if  it  rose  only  to  the  harmonic  mean,  the 
mean  exchange-value  of  the  two  things  would  fall  to  the  arith- 
metic mean,  and  a  similar  inconsistency  would  be  found. 

The  error  in  the  position  of  the  arithmetic  averagist  is  evi- 
dent. The  higher  A  rises  from  its  original  position,  the  smaller 
is  its  rise  in  teach  stag-e  of  its  advance.  Yet  the  arithmetic  aver- 
agist  accords  to  it  the  same  influence  upon  the  mean  when  it 
rises  from  1.99  to  2.00  as  when  it  rose  from  1.00  to  1.01,  al- 
though the  mean  has  lagged  behind  somewhere  below  1.50, 
where  its  rise  by  half  of  one  per  cent,  at  a  time  (reckoned  in 
1.00)  is  considerably  more  than  half  as  large  as  the  rise  of  A 
from  1.99  to  2.00.  And  reversely,  when  A  alone  falls,  while 
it  is  fiilling  from  .02  to  .01,  the  arithmetic  averagist  accords  to, 
it  not  so  nnich  inlluence  upon  the  mean  as  when  it  fell  from 
1.00  to  .99,  although  its  fall  is  now  fifty  times  greater  than  it 
was  then. 

But  the  geometric  averagist,  in  placing  the  mean  at  1.41 
when  A  alone  rises  from  1.00  to  2.00,  ])laces  it  where  the  in- 
fluence of  the  later  rises  is  the  same  as  that  of  the  earlier,  when 
in  the  same  percentage.  Thus  if  A  alone  rises  first  to  1.41  by 
41  per  cent.,  the  mean  is  placed  at  1.19,  indicating  a  rise  by 
19  per  cent.  Then  wiien  A  rises  further  to  2.00,  this  also  be- 
ing a  rise  b)'  41  })er  cent.,  the  same  influence  as  before  is  attrib- 


AIM'MKD    TO    EXC1IAXGE-VAI.UES  255 

uted  to  it  ;  for  the  rise  of  the  inciin  from  1.19  to  1.41  is  by  19 
per  cent. 

§  4.  Here  we  might  pause,  our  labor  done,  but  for  tlie  fact, 
above  shown,  that  what  is  true  of  the  geometric  mean  is  not  true 
of  tlie  geometric  average.  All  that  has  just  been  demonstrated 
has  been  demonstrated  on  the  supposition  that  we  are  dealing 
with  only  two  classes  of  things,  and  these  equally  large  or  im- 
portant over  both  the  periods  compared.  If  the  two  classes  are 
not  equally  large  over  both  the  periods,  or  if  there  are  many 
varying  classes  to  be  considered,  nothing  that  has  just  been 
proved  applies, — nor  does  it  apply  with  accuracy  if  we  use  the 
geometric  average  with  the  proi3er  weighting  over  bofli  the 
periods.  For  instance,  suppose  the  classes  are  equally  impor- 
tant only  at  the  first  period,  and  as  [A]  rises  in  price  it  rises 
also  in  importance,  and  as  [B]  falls  in  price  it  falls  also  in  im- 
portance. Then  it  is  evident  that  a  rise  of  A  from  1.00  to  1.50 
is  not  fully  compensated  by  the  fall  of  B  to  the  other  geometric 
term,  O.GGf  ;  but,  for  compensation,  B  must  fall  further.  How 
much  further  it  must  fall  will  depend  upon  the  extent  of  the  al- 
teration in  the  relative  importance  of  the  classes.  It  is  possible, 
therefore,  that  the  price  of  B  may  have  to  fall  to  the  arithmetic 
term,  0.50.  We  shall  in  fact  find  this  to  be  the  case  when  the 
relative  sizes  of  the  classes  vary  exactly  as  their  prices.  Yet 
this  will  not  be  exactly  indicated  by  the  geometric  average  with 
the  proper  (uneven)  weighting  for  both  the  periods. 

Therefore  we  need  to  turn  to  the  examination  of  the  averages 
more  closely  in  connection  with  the  subject  of  weighting.  Also 
it  is  always  well  to  examine  the  arguments  of  persons  who  have 
advocated  other  opinions,  and  although  we  have  already  dis- 
proved the  positions  of  the  arithmetic  and  harmonic  averagists 
under  certain  provisos,  yet  their  general  position  still  remains. 
It  happens  that  both  these  objects  may  be  pursued  together. 


CHAPTER    IX. 

REVIEW  OF  THE    ARGUMENTS  FOR  THE  HARMONIC  AND 
ARITHMETIC  AVERAGES  OF  PRICE  VARIATIONS. 


§  1.  The  ar<>'iimeiits  for  the  harmonic  and  arithmetic  averages 
have  been  very  imperfectly  stated.  We  shall  therefore  have  to 
try  to  understand  them  not  merely  as  they  have  been  presented, 
but  as  they  turn  out  to  be  on  fuller  analysis.  They  have  gen- 
erally been  adduced  in  the  form  which  considers  what  constitutes 
variation.  But  this  depends  upon  what  constitutes  constancy. 
Also  little  or  no  reference  has  been  made  to  weighting.  This 
we  shall  have  to  supply.  We  may,  however,  supply  it  later, 
first  reviewino-  the  aro-uments  in  their  most  o-eneral  forms. 

In  the  argument  for  the  use  of  the  harmonic  average  of  price 
variations  tiie  idea,  when  there  is  supposed  to  be  constancy,  is 
as  follows.  The  same  total  sum  of  money  purchasing  the  same 
total  quantity  of  all  kinds  of  goods  at  both  the  periods  com- 
pared, the  purchasing  power  of  this  sum  (and  consequently  the 
exchange-value  of  money)  is  considered  not  to  Ikinc  varied,  what- 
ever be  the  c^hangcs  in  the  make-uj)  of  the  total  (piantity  of 
goods,  an  increase  in  the  quantity  of  one  class  of  things  pur- 
chasable with  th(!  same  ])arti('ular  sum  of  money  devoted  to  pur- 
chasing it  at  both  the  periods  being  otfset  by  an  equal  decrease 
in  the  quantity  of  another  class  of  things  purchasable  with  the 
particular  sum  devoted  to  purchasing  it  at  the  two  periods. 
For  the  total  of  the  (luantities  remains  the  same  when  tliese 
quantities  vary  oppositely  to  the  arithmc^tic  terms,  and  conse- 
quently wiien  their  prices  vary  oppositely  to  the  iiarmonic  terms  ; 
so  that  if  the  harmonic  average  of  the  |)rice  variations  indicates 

256 


ANALYSIS    OF    TlfH    A  !.'( ;  I'M  r;.\TS  257 

constancy,  it  indicates  tin's  condition.  As  the  <|nantities  of"  tlie 
goods  are  in  aritlinietic  progi'cssion,  the  comi)ensation  may  be 
described  as  arltliiiiefir  eompotxation  by  equal  mass-quantities — 
a  loss  of  one  third  on  A,  for  instance,  being  compensated  by  a 
gain  of  one  third  on  B,  or  by  a  gain  of  one  sixth  both  on  B  and 
on  C.  Now  if  a  variation  occurs  in  the  total  qnantity  of  the  goods 
purchasable  at  the  second  period  from  that  at  the  first,  this 
variation  of  the  total  is  indicated  by  the  arithmetic  average  of 
the  variations  of  the  particular  quantities ;  f  )r  if  the  ({uantities 
purchasable  all  varied  at  the  same  common  average  rate,  the 
same  result  in  the  total  would  be  obtained.  Tluis  if  there  is  a 
loss  of  one  third  in  tlie  (piantity  of  A  alone,  this  is  the  same, 
among  two  classes  of  commodities,  as  a  loss  of  one  sixth  on  each  ; 
among  three  classes,  of  one  ninth  on  each  ;  and  so  on,  just  the 
same  as  if  each  of  the  two  classes  had  lost  one  sixth,  or  each  of 
the  three  classes  one  ninth,  and  so  on.'  Hence  this  same  result  is 
obtained,  for  the  prices,  by  the  harmonic  average  of  the  price  vari- 
ations. All  this  is  a  mathematical  fact.  Upon  this  fact  is  based 
the  argument,  if  argument  it  may  be  called,  or  rather  the  claim, 
suggested  by  Jevons  and  insisted  upon  by  Messedaglia,  that  the 
harmonic  average  of  the  price  variations,  because  it  inversely 
indicates  this  variation,  or  the  preceding  constancy,  of  the  total 
quantity  of  goods  purchasable  with  the  same  total  sum  of  money, 
also  inversely  indicates  the  variation,  or  constancy,  of  the  purchas- 
ing power  of  money  (and  consequently  of  its  exchange-value), 
and  so  is  to  be  taken  as  the  proper  method  of  measuring  it. 

In  this  argument  it  is  always  understood  that  the  particular 
sums  of  money  spent  on  every  class  of  goods  remains  constant 
at  both  the  periods,  whatever  be  the  variations  in  the  })articular 
mass-quantities  therewith  purchased. 

§  2.  In  the  argument  for  the  use  of  the  arithmetic  average  of 
price  variations  the  idea,  when  there  is  supposed  to  be  constancy, 
is  as  follows.  The  same  total  quantity  of  goods  l)eing  purchas- 
able with  the  same  total  sum  of  money  at  both  the  ])eriods,  a 
total  price  is  conceived  of  the  total  (piantity,  whicli  total   price 

1  This  will  he  recognized  as  the  argument  we  lighteil  upon  at  tirst  glance  ahove 
in  Chapter  VI.  Sect.  I.  '^/,2. 

17 


258         TiiK  Ai!(.r.\ii:N'i>>   i-ou  thk  othki;  averages 

1ms  not  varied  wliatever  be  the  changes  in  its  make-up,  that  is, 
in  the  [)artic'ular  prices,  an  increase  in  the  price  of  one  class  of 
things  being  offset  by  an  e({ual  decrease  in  the  price  of  another. 
For  the  total  of  the  j)rices  remains  the  same  when  they  vary 
<)})j)ositely  to  the  arithmetic  terms  (and  consequently  when  the 
mass-quantities  purchasable  vary  oppositely  to  the  harmonic 
terms) ;  so  that  if  the  arithmetic  average  of  the  price  variations 
indicates  constancv,  it  indicates  this  condition.  And  so  there  is 
arithiiietic  voinpensdtion  hi/  equal  .sniiii^  of  money — a  need,  for  in- 
stance, foi-  one  half  more  money  to  purchase  A  being  offset  by  a 
need  for  one  half  less  money  to  purchase  B.  Now  if  a  varia- 
tion ()C(!urs  in  the  total  pricej  or  total  sum  of  money  needed  to 
purchase  the  same  quantities  of  the  same  classes  of  goods  at  the 
second  period  from  what  was  needed  at  the  first,  this  variation 
of  the  total  is  indicated  by  the  arithmetic  average  of  the  partic- 
ular prices  ;  for  if  the  prices  all  varied  at  the  same  common 
average  rate,  the  same  result  in  the  total  price  would  be  ob- 
tained. Thus  a  rise  in  the  price  of  one  class  of  commodities  by 
fifty  per  cent,  has  the  same  influence  upon  the  total  as  a  rise  of 
two  classes  by  twenty  five  per  cent.,  of  three  classes  by  sixteen 
and  two  thirds  per  cent.,  and  so  on.  Hence  the  result  in  the 
variation  of  the  total  price  is  obtained  by  the  arithmetic  average 
of  the  variations  of  the  particular  j)rices.  Here  again  is  a 
mathematical  fact,  upon  which  is  based  the  argument,  or  claim, 
that  the  arithmetic  average  of  the  price  variations,  because  it 
dire(rtly  indicates  this  variation,  or  the  preceding  constancy,  of 
the  total  price  (tf  the  same  goods,  also  inversely  indicates  the 
variation,  or  eonstaney,  in  tiie  pui'ehasiiig  power  of  money  (and 
consequently  in  its  exchange-value)  and  so  is  to  be  taken  as  the 
])roper  method  of  measuring  it. 

In  this  ai'gument  it  is  always  understood  that  the  particular 
quantities  purchased  of  e\-ery  class  remain  constant  at  both  the 
periods,  whatever  be  the  variations  in  the  particular  sums  of 
money  needed  to  purchase  them. 

$  I).  Thus  the  harmonic  and  aritlunetie  avei-ages  of  prices,  in 
the  minds  of  their  advocates,  represent  reversed  positions.  The 
former  is  directh"  applie(l  to  the  measurement  of  the  pun^hasitig 


ANALYSIS    OF     rilK    AlKil'M  HNTS  259 

power  (»f  money  by  the  total  qiiaiitity  of  all  coniniodities  a  ^iven 
total  sum  of  money,  spent  in  the  same  way  at  both  ])eri<)(ls,  will 
purchase.  The  latter  is  applied  rather  to  the  measurement  of 
the  power  of  all  commodities  over  money  by  the  total  sum  of 
monev  a  total  quantity  of  ^ivcn  ]iarticular  conunodities,  (•omj)osed 
in  the  same  way  at  both  ])eriods,  will  command  in  exchange. 
The  former  makes  use  of  arithmetic  compensation  by  ccjual  mass- 
quantities,  the  latter  of  arithmetic  compensation  by  equal  sums 
of  money.  They  have  in  common  the  use  of  arithmetic  com- 
})ensation.  They  dilfer  in  applying  this  to  opposite  sides  of  the 
question,  in  each  case  excluding  notice  of  the  other  side. 

When  we  look  at  either  of  these  positions  by  itself,  it  seems 
verv  stroup-.  The  one  inverselv  measures  variations  in  the  total 
mass-quantity  purchasable  with  a  given  sum  of  money.  Does 
it  not  then  inversely  measure  variations  in  the  purchasing  })Ower 
or  exchange-value  of  money,  and  directly  measure  variations  in 
the  general  level  of  prices  ?  The  (^tlier  directly  measures  varia- 
tions in  the  total  price  of  given  quantities  of  all  things  together. 
Does  it  not  then  directly  measure  variations  in  the  general  level 
of  prices,  and  inversely  variations  in  the  purchasing  ])ower  or 
exchange-value  of  money  ? 

Each  of  these  methods  is  founded  on  a  procedure  ^vhich  we 
employ  with  regard  to  single  classes  of  commodities.  We 
measure  the  coustimcy  or  variation  of  the  particular  exchange- 
value  of  money  in  any  one  class  of  commodities  by  the  constancy 
or  variation  of  the  quantity  of  this  commodity  jiurchasable  ^vith 
a  given  sum  of  money  at  each  of  the  periods  (according  to  Prop- 
osition L), — and  by  in^•erting  the  result  so  obtained  we  can 
measure  the  constancy  or  variation  of  the  price  of  this  thing  (al- 
though we  never  adopt  this  roundabout  course).  Hereupon  the 
harmonic  averagist  of  prices  concludes  that  we  can  measure  the 
constancy  or  variation  of  the  general  exchange- value  of  money 
in  all  tilings  by  the  constancy  or  variation  of  the  total  ipiantity 
of  things  purchasable  at  each  of  the  periods  with  a  given  total 
sum  of  money  (which  he  further  specifies  must  be  spent  in  the 
same  way  at  both  the  periods), — and  the  inverse  of  the  result  so 
obtained  he  regards  as  the  proper  measure  of  the  constancy  or 


2()0  THE    AHCiUMENTS    FOR    THE    OTIIKI;    AVKI;A(;ES 

variation  of  prices  in  general.  .Vgain  we  mea.<ure  the  eon.staney 
or  variation  of  the  particular  price  of  any  one  class  of  commod- 
ities by  the  constanc}^  or  variation  of  the  price  of  a  given  quan- 
tity of  this  commodity  at  each  of  the  ])eriods, — and  the  inverse 
of  this  gives  the  constancy  or  \ariati(tn  of  the  particular  ex- 
change-value (A'  moniy  in  that  class  (according  to  Proposition 
X.).  Hereupon  the  arithmetic  avcragist  of  prices  concludes 
that  we  can  measure  the  constancy  or  variation  of  a  total  price 
of  all  things  by  the  constancy  or  variation  of  the  total  price  at 
each  period  of  a  given  total  (piantity  of  all  things  (further  s])eei- 
fying  that  these  quantities  must  be  individually  the  same  at 
both  periods), — and  the  inverse  of  the  result  so  obtained  (the  in- 
verse of  the  constancy  or  variation  of  this  total  price  of  all  the 
same  things)  he  regards  as  the  proper  measure  of  the  constancy 
or  variation  of  the  general  exchange-value  of  money  in  all  things. 

In  the  two  particular  measurements  that  serve  as  models  for 
these  two  general  measurements  there  is  no  disagreement  pos- 
sible. The  results  obtained  by  the  one  are  universally  the  same 
as  the  results  obtained  by  the  other.  Those  two  particular  meas- 
urements are  also  always  applicable  to  every  one  and  the  same 
case.  The  two  copies,  however,  are  really  applicable  each  to  a 
different  state  of  things,  so  that  they  arc  not  properly  contra- 
dictory or  antagonistic.  Yet  this  fact  has  been  mostly  over- 
looked, and  the  advocates  of  these  different  averages  have  simply 
urged  the  use  of  the  one  or  of  the  other  method  for  all  cases. 
When  applied  to  the  same  cases,  differently  garbling  them  to  fit 
them  to  the  different  re(j[uirements,  these  methods  give  different 
results.  This  disajrreement  shows  that  at  least  one  of  them  is 
false,  and  ])r()l)ably  both,  when  ap])lied  in  such  a  loose  way. 
But  confined  each  to  the  cases  that  may  hap})en  to  exist  to  which 
it  is  applicable,  they  may  botli  be  true,  or  they  may  l)oth  be  false, 
or  the  one  may  be  true  and  the  other  false.  ^Vt  all  events  this 
divergence  of  the  copies  from  the  models  shows  that  something 
is  wrong  in  the  co})ying  at  least  in  one  case. 

^  4.  The  faidtiness  of  the  e()])y  in  the  positifni  of  the  harinouic 
avcragist  of  prices  is  not  far  to  seeU.  in  measuring  the  con- 
stancy or  variati()n  of  the  particular  exehange-value  or  purchas- 


ANAJ.VSIS    OF    THE    AlKJUMENTS  261 

ing  })()\vci'  (jf  inoncy  in  any  cla.s.s  of  eoramoditics  by  tlie  ('onstancy 
or  vai'iatioii  in  tiie  qnantity  of  it  purchasable  at  each  period  with 
a  given  sum  of  money,  we  are  careful  to  note  that  the  quality  of 
this  mass-cjuantity  must  not  change,  and  guarding  this,  there  is 
no  possibility  of  divergence  in  our  result,  no  matter  what  mass- 
unit  we  use.  Now  when  Ave  try  to  imitate  this  operation  in 
measuring  the  constancy  or  variation  in  the  general  exchange- 
value  or  purchasing  power  of  money  in  many  or  all  classes  of 
commodities,  we  at  once  strike  upon  a  difficulty  if  there  is  the 
least  change  in  the  relative  exchange-values  or  prices  or  pre- 
ciousness  of  these  classes.  For  if  there  is  any  such  change, 
this  is  the  same  as  a  change  in  the  quality  of  the  total  mass- 
quantity,  which  renders  the  comparison  of  its  change  of  size 
nugatory,  unless  it  can  be  allowed  for.  Furthermore,  the  com- 
parison of  the  total  mass-quantities  purchasable,  or  actually  pur- 
chased, at  each  period  with  the  same  sums  of  money,  will  now 
be  different  according  to  the  mass-units  that  are  used  in  each 
class.  The  harmonic  averagist  of  ])rices  adopts  the  usual  prac- 
tice in  these  matters  and  takes  as  his  mass-unit  in  every  class  the 
mass  that  is  equivalent  to  the  money-unit  at  the  first  period. 
But  this  is  only  one  of  many  possible  ways  of  selecting  the 
mass-units,  and  he  has  offered  no  reason  for  adopting  it — and 
apj)arently  has  none,  except  the  blind  following  of  a  convenient 
hal)it.  Doing  so,  when  he  finds  the  mass-quantities  purchasable 
with  the  same  sums  at  each  period  to  foot  np  to  the  same  total 
quantity,  so  that  he  concludes  that  the  exchange-value  of  money 
has  not  altered  on  the  whole,  it  may  be  that  the  total  weight,  or 
the  total  bulk,  of  the  goods  so  purchasable  with  the  same  sums 
of  money  may  be  very  different  at  the  two  periods.  Hence  he 
is  here  not  following  his  model.  And  if  he  should  now  try  to 
return  to  his  model,  he  would  be  confronted  with  the  question. 
Is  he  to  require  the  same  total  weight,  or  the  same  total  bulk  ? 
Between  these  two  there  is  nothing  to  decide.  But  if  he  does 
(arl)itrarily)  hit  upon  the  one  or  the  other,  he  will  only  be  meas- 
uring the  constancy  or  variation  in  the  one  or  in  the  other  kind 
of  preciousness  of  the  goods  relatively  to  money."     Thus  the 

-  He  woulil  he  ii«iiig  Drobiscli's  method  applied  to  cases  in  whicli  XiUi  =  .roa2 


2<)"i  iiii;   Ai;(;r>fKN  rs   vou    riii-:  oriiKi;   a\i:i;a(;ks 

incasurt'inent  ol'  the  li'enoral  ('X('luiiiu('-\:iliu'  of"  inoiicy  rannot 
he  so  smiply  made  <>n  the  model  of  this  method  of  measuring  a 
particular  exchauge-value  of  money.  The  imitation  will  be  ex- 
act, in  case  the  total  mass-quantities  are  the  same  at  both  periods, 
only  if  there  are  no  ])riee  variations,  and,  in  case  the  total  mass- 
(piantities  are  different,  only  if  all  the  j)riees  have  varied  in  the 
same  pro])ortion — that  is,  only  when  any  other  averatie  wonld 
be  as  good.  There  is,  however,  a  more  complex  May  in  which  this 
model  may  be  imitated.  But  the  harmonic  averagist  has  not 
sought  it,  and  nobody  has  hitherto  pointed  it  out. 

In  the  argument  for  the  arithmetic  average  of  prices  there 
is  a  somewhat  similar  defect.  But  this  seems  to  bear  with  it  its 
own  correction,  in  the  way  alluded  to  at  the  end  of  the  preced- 
ing Chapter.  Here,  m  its  totality  and  in  its  details,  identically 
the  same  (or  similar)  mass  of  goods  is  used,  of  which  the  con- 
stancy or  the  variation  in  the  total  price  is  taken  as  the  measure 
of  the  constancy  or  inverse  variation  in  the  exchange-value  of 
money.  ^V^nd  consequeutly  there  is  no  difficulty  here  about  the 
mass-units  to  be  em])loyed.  But  this  total  mass  is  not  econom- 
ically the  same  in  all  its  parts  at  both  periods,  unless  all  prices 
have  remained  constant  or  varied  alike  ;  for  otherwise  some  of 
its  parts  have  become  more  or  less  precious,  and  also  more  or 
less  important,  than  others,  and  so  its  economic  make-up  has 
altered.  Still  it  is  precisely  the  prices  of  the  things  that  have 
grown  more  precicjus  and  more  important  that  have  risen,  and 
the  prices  of  the  things  that  have  become  less  precious  and  less 
important  that  have  fallen,  and  all  these  variations  are  in  ex- 
actly the  same  proportions  ;  which  is  about  as  we  should  desire. 
Hence  it  is  possible  that  this  measurement  is  a  good  copy  of  its 
nutdel.  Bnt  it  would  [)rol)ably  be  more  difficult  to  prove  this 
than  to  ])rove  the  correctness  of  the  arithmetic  average  (in  the 
special  cases  to  which  it  is  claimed  to  be  a|)])licable).  Hence 
the  above  argument  made  by  the  arithmetic  avcragists  is  still 
unsatisfactory,  and  we  must  continue  to  ])robe  it. 

l/i^i  =  1/2P2J  (because  the  same   suras  are  supposed  to  be  spent  on  every 

class  at  both  periods),  so  that  the  first  part  of  I)n)l)iscirs  formula  falls  away, 
being  reduced  to  unity. 


MATIIKMA  TKAI.     i;  III  ,A  TlONS     K  ;  .\<  )i;  Kl )  "if)."} 

ir. 

§  1.  Both  the  above  ar<jciiinents  have  ))eeii  advanced  only  on 
very  cursory  inspection  and  inconi|)lete  analysis  of  tlie  mathe- 
matical relations  involved.  The  writers  ac(^uainted  with  them 
seem  to  have  thonght  that  tlie  argument  for  tlie  harmonic  aver- 
age is  ])eculiar  to  that  average,  a])])licable  neither  to  the  arith- 
metic nor  to  the  geometric,  and  that  the  argument  for  the  arith- 
metic average  is  peculiar  to  that  average,  applicable  neither  to 
the  harmonic  nor  to  the  geometric.  And  as  these  two  argu- 
ments, attacking  our  problem  from  its  two  opposite  sides,  a])])ear 
to  occu])y  all  the  possil)le  positions,  it  has  seemed  as  if  no  room 
were  left  for  the  geometric  average  or  mean.  This  lias  seemed 
to  stand  out  in  the  cold,  with  no  function  to  fulfil,  and  with  no 
argument  ajjplicable  to  it.  Hence  the  neglect  with  which  it  has 
been  treated.  Yet  not  much  analysis  is  needed  to  show  that 
these  views  are  false. 

The  argument  for  the  harmonic  average  of  prices  assumes  a 
certain  distribution  of  our  spendings,  and  then  finds  compensa- 
tion arithmetically  by  equal  mass-quantities,  so  that,  when  con- 
stancy is  indicated,  an  equal  total  mass-quantity,  though  differ- 
ently made  up,  is  purchasable  at  both  periods  with  the  same 
sura  spent  in  the  same  ^va}•  at  both  pei'iods.  Now  suppose 
prices  have  changed  to  the  arithmetic  extremes,  namely  in  our 
example  from  1.00  to  l.oO  and  from  1.00  to  .50.  Then  if  we 
spend  1.50  on  [A]  at  both  periods  and  .50  on  [B]  at  both 
periods,  we  get  for  2.00  at  the  first  period  1|  A  and  |  B,  and 
at  the  second  period  1  A  and  1  B, — losing  ^  A  and  gaining  |  B. 
Thus  there  is  compensation  by  equal  mass-quantities,  and  ability 
with  the  same  sum  to  purchase  an  equal  total  mass-quantity. 

The  argument  for  the  arithmetic  average  of  prices  assumes  a 
certain  distribution  of  our  purchases,  and  then  finds  compensa- 
tion arithmetically  by  equal  sums,  so  that,  when  constancy  is 
indicated,  the  same  total  sum,  though  difFerently  made  up,  is 
able  to  purchase  at  both  j>eriods  the  same  total  mass-quantity 
made  up  of  the  same  particular  mass-quantities  at  both  periods. 
Now  suppose  prices  have  changed  to  the  harmonic  terms,  namely 


2(54         riiK  Ai{(;r>rENTs  foi;  tiik  othkk   avi;i;a(;ks 

ill  our  t'xanij)lc  fVoin  1.00  to  l.oO  :iinl  tVoin  IJM)  to  .To.  Tlu'ii 
if  wo  j)uroha.se  |  A  and  1^  A  at  both  periods,  we  can  do  so  at 
the  first  j^eriod  by  sjx'nding  .GOg  on  [A]  and  l.."v'ij  on  [B]j 
and  at  the  second  period  by  sjiendinjr  1.00  on  [A]  and  1.00  on 
[B]— or  .'.  M  more  oil  [A]  and  I,  M  U'ss  on  [H].  Thus  there  is 
compensation  by  ecpial  sums,  and  ability  to  ])urehasc  exactly  the 
same  i)articular  niass-(piantities  with  the  same  total  sum. 

Suppose,  ag:ain,  that  ]>rices  have  ehanued  to  the  o:eometric 
terms,  namely  from  1.00  to  l.oO  and  from  1.00  to  .<!()§.  Then 
if  we  sjx'ud  1.20  on  [A]  at  both  periods  and  .SO  on  [B]  at 
both  periods,  we  uet  for  2.00  at  the  first  ])eriod  1^  A  and  |  B 
and  at  the  second  period  j^  A  and  li  I>, — losing  ^  A  and  gain- 
ing I  I>.  Thus  there  is  com})ensation  by  equal  mass-quantities, 
and  ability  with  the  same  sum  to  get  an  ecpial  total  mass-quan- 
tity. Also  if  we  [)urchase  ^  A  and  l^-  B  at  both  periods,  we 
can  do  so  at  the  first  period  by  spending  .80  (»n  [A]  and  1.20 
(»n  [B]  and  at  the  second  j)eriod  by  spending  1.20  im  [A]  and 
.SO  on  [B]— or  I  M  more  on  [A]  and  f  M  less  on  [B] .  Thus 
there  is  compensation  by  equal  sums,  and  ability  to  purchase 
the  same  ])ai'tieular  mass-quantities  Avith  the  same  total  sum. 

Hence  the  argument  which  seems  to  be  [)eculiar  to  the  har- 
monic average  a])plies  also  to  the  arithmetic  and  to  the  geo- 
metric ;  and  the  argument  which  seems  to  be  peculiar  to  the 
arithmetic,  applies  also  to  the  harmonic  and  to  the  geometric. 
^And  the  geometric  average,  or  at  least  the  geometric  mean,  in- 
stead of  being  without  any  argument,  and  without  any  function, 
equally  is  subject  to  either  of  the  arguments,  and  e(pially  per- 
forms the  same  functions.  Instead  of  standing  out  in  the  cold, 
it  belongs  in  the  fold  ;  and  we  cannot  examine  either  of  the  ar- 
guments far  witiiout  taking  also  it  into  consideration. 

Thus  i\w  there  two  arguments,  in  the  forms  hitherto  used,  for 
three  nieiuis  or  averages,  each  argument  being  found  to  be  appli- 
cable t^)  each  of  the  means.  Therefore,  so  far  as  we  yet  see, 
any  one  mean  can  ap|)ar('iitly  be  argued  for  as  well  as  any  other 
by  either  of  these  arguments.' 

*  That  there  is  ability  to  purchase  with  the  same  sum  at  both  periods  an  equal 
total  (luantity  of  commodities  differently  made  up  [hence  with  compensation  by 


MATIIKMATK  AL    RELATION'S    IGNORP^D  265 

§  2.   FurtluTinorc,  wIkmi   jji-iccs  change  to   the  harmonic  ex- 
tremes and  the  harmonic  average  indicates  constancy,  or  when 
prices  change  to  the  arithmetic  extremes  and  the  harmonic  aver- 
age indicates  constancy,  or  when  jn'iccs  change  to  the  geometric 
extremes  and  tlie  geometric  mean  indicates  constancy,  although  it 
is  possible  in  each  case  with  the  same  sum  of  money  spent  in  the 
same  way  at  both  periods  to  purchase  an  equal  total  ([uantity  of 
g(»ods  differently  made  up,  the  spendings  and  the  purchases  be- 
ing different  in  each  of  the  supposed  changes,  it  is  also  possible 
in  each  case  with  the  same  sum  of  money  spent  in  the  same  way 
at  both  ])eriods  to  get  both  a  larger  and  a  smaller  total  quantity  of 
goods  at  the  second  than  at  the  tirst  j)eriod,  our  spendings  being 
variously  distributed.     Thus    in    our  simple    example  of  two 
classes  of  commodities  both  priced  at  1 .00  at  the  first  period  we 
can  at  that   period  itself  with  2  M  purchase   1  A  and    1  B,  or 
two  given  quantities  of  [A]  and  [B]  together,  and  in  any  com- 
bination of  s])endings  of  the  same  sum  we  always  can  purchase 
two  such  quantities — e.  g.,  |  A  and  1^  B,  |  A  and  1|  B,  li  A 
and   I  B,  2  A  and  0  B,  0  A  and   2  B,  etc.     But  at  the  second 
period  with  prices  changed  to  the  harmonic  terms,  1.50  and  .75, 
if  we  employ  2  ]VI  to  purchase  1  A  and  some  B,  ^^^e  must  use 
H  of  it  in  purchasing  1  A  and  have  left  only  \  with  which  we 
can  purchase  only  |  B,  or  all  told  only  If  of  [A]  and  [B]  to- 
gether ;  or  if  we  purchase  1  B  with  |  M,  with  the  remaining  1^ 
M  we  can  purchase  only  |A,  or  1|  of  [A]  and    [B]  together 
— in  l)oth  cases  a  smaller  total  than  before.     Or  again,  employ- 
ing 2  M  to  purchase  |  A  and  some  B,  we  can  purchase  If  B, 
or  together  2^,  this  time  more  than  before.     With  the  prices 
changed  to  the  arithmetic  terms,  1.50  and  .50,  by  employing 
1  ^I  to  purchase  some  A  and  1  M  to  purchase  some  B,  we  get 
I  A  and  2  B,  or  on  the  whole  more  than  before ;  or  by.  spend- 
ing 1|M  on  [A]    and  ^M  on    [B]  we  get  li  A  and  ^B,  or 
on  the  whole  less  than  before.     And  lastly  if  the  prices  change 
to  the  geometric  terms,  1.50   and  .66|,  we  can  also  get  some- 
times more  and  sometimes  less  of  [A]  and  [B]  together — more, 

equal  mass-quantities]  even  when  the  simple  harmonic  average  of  the  price  varia- 
tions indicates  constancy,  was  perceived  by  Walras,  B.  69,  pp.  14-15.  But  Walras 
has  not  investigated  further. 


2li(i  THK    ARGUMENTS    Foi;    THE    OTHKI!    AVEHA(;ES 

for  instance,  if  we  spend  1  M  on  [A]  and  1  M  on  [B],  gettinir 
I  A  and  U  B,  or  all  told  2^,  or  less  if  we  spend  If  M  on  [A] 
and  i  M  on  [B],  ,<Tettino-  1^  A  and  I  B,  or  all  told  li|. 

Henee,  if  the  mere  j)ossibility  of  oettino-,  with  onr  money 
s|)ent  in  the  same  way  at  l)oth  periods,  an  ('(pial  total  (juantitv 
of  connnodities  is  a  reason  for  thinking;  onr  money  constant  in 
purchasing  power,  the  .simnltaneons  possibilit\'  of  getting  both 
more  and  less  is  an  equally  good  reason,  so  far  as  is  yet  shown 
to  the  contrary,  in  each  case,  to  think  that  our  money  has  l)oth 
a])preeiated  and  depreciated  ;  which  is  absurd. 

-Vgain,  although  it  is  possible  in  each  of  these  cases  with  the 
same  total  sum  of  money  differently  spent  to  purchase  at  both 
periods  exactly  the  same  quantities  of  commodities,  these  being 
different  in  each  of  the  supposed  cases,  it  is  also  possible  in  each 
case  that  to  purchase  other  exactly  the  same  quantities  of  com- 
modities, a  larger  or  smaller  sum  of  money  is  needed  at  the  sec- 
ond ])eriod.  This  could  be  easily  shown  in  our  simj)le  example. 
But  enough  has  been  shown  already. 

Hence,  if  the  mere  possibility  of  getting  at  l)oth  periods  ex- 
actly the  same  quantities  of  commodities  with  the  same  total 
sum  of  money  is  a  reason  for  thinking  the  level  of  jn'ices  con- 
stant, the  simultaneous  possibility  that  for  getting  exactly  the 
same  other  quantities  of  commodities  both  a  larger  and  a  smaller 
total  sum  may  be  needed,  is  equally  good  reason,  so  far  as  is 
yet  shown  to  the  contrary,  in  each  case  to  think  that  the  level 
of  prices  has  both  risen  and  fallen  ;   which  again  is  absurd. 

Therefore,  so  far  as  we  yet  see,  these  arguments  ap|)arently 
arc  equally  defective  whether  a])|)lied  to  the  liarnionic,  aritli- 
metic  or  geometric  averages  or  metuis. 

§  3.  This  defectiveness  of  the  arguments  seems  to  have  been 
ignored.  The  conflicting  possibilities  have  been  overlooked. 
Hardly  any  advance  has  been  made  in  this  matter  since  the 
famous  dispute  between  .Tev(ms  and  fjaspeyres.  That  dispute, 
therefore,  deserves  review  ;  i'or  it  may  |»rovi<le  a  warning,  still 
needed  in  our  subject,  against  reasoning  which  stop^  at  a  few 
half-truths  first  lighted  upon. 

Jevons  had  noticed    in  his  first  work  that  the  price  of  cocoa 


MATHEMATICAL    RELATIONS    KJNORED  267 

had  recently  risen  100  per  cent.,  while  that  of  cloves  had  fallen 
50  per  cent.,  and  had  said  it  wonld  be  "  totally  erroneous  "  to 
say  the  average  change  was  a  rise  of  25  per  cent.,  since  the  geo- 
metric mean  in  this  case  indicates  no  variation  at  all.'  Here- 
upon Laspeyres  commented  and  argued  as  follows: — "The 
geometric  mean  expresses  neither  the  depreciation  of  commodi- 
ties or  appreciation  of  money,  nor  the  appreciation  of  commodi- 
ties or  de])reciation  of  money — that  is,  according  to  Jevons, 
increase  or  decrease  in  its  '  potency  in  purchasing  other  articles.' 
Let  us  retain  the  example  used  by  Jevons.  Here,  after  the 
change  in  price  of  cocoa  and  cloves,  the  same  sum  of  money  has 
not  the  same  purchasing  power  as  before,  but  a  smaller  one,  and 
exactly  so  much  smaller  as  is  indicated  by  the  arithmetic  mean. 
If  a  certain  weight  of  cocoa  (say  1  cwt.)  previously  cost  100 
thalers,  and  a  certiun  weight  of  cloves  (say  1  cwt.)  also  cost  100 
thalers,  and  the  price  of  this  amount  of  cocoa  rises  from  100  to 
200  thalers,  and  that  of  the  cloves  falls  from  100  to  50,  then  200 
thalers  no  longer  have  the  same  potency  in  })urchasing  cocoa 
and  cloves.  For  this  sum  the  ]>urchaser  })rocures  only  |  cwt. 
cocoa  (=  150  th.)  and  1  cwt.  cloves  (=  50  th.),  or  he  procures 
1  cwt.  cocoa  (=  200  th.)  and  no  cloves  at  all.  The  purchasing 
power  is  now  ^  less,  that  is,  the  ])urchaser  must  add  ^  in  order 
to  get  the  same  quantity  ;  or  the  250  thalers  are  now  by  ^  (50 
th.)  less  worth  than  formerly.      Exactly  this  is  ex])ressed  by  the 

•200  +  50 
arithmetic  mean =  125  ;    125   thalers   liave  only  tlie 

same  jjurchasing  power  as  100  before,  or  250  only  the  same  as 
200  before.  Money  has  depreciated  20  per  cent.;  commodities 
have  risen  25  per  cent.  What  is  true  of  tlie  average  of  two 
commodities,  is  true  also  for  any  nnml)cr  of  commodities."  ^ 

Laspeyres  thus  found  fault  with  the  geometric  average  f(jr  in- 
dicating constancy  in  the  "  potency  in  purchasing  "  under  the 
given  conditions,  because  these  conditions  permit  us  with  one 
whole  sum,  200  thalers,  to  get  at  the  second  period  only  1  cwt. 
cocoa  and  no  cloves,  that  is,  a  smaller  quantity  than   before, 

2  B.  22,  pp.  23-24. 
'  B.  25,  p.  97. 


2f)S  Ai;<;rMKNTs  for  the  othki:  aveeages 

which  tact  he  took  for  an  indication  tliat  tlic  '^  potency  in  \mv- 
chasinu"  was  smaller;  to  which  came  the  added  evidence  that 
more  monev  is  required  at  tlie  second  period  to  buy  the  1  cwt. 
of  each  article.  He  omitted  to  state  that  these  conditions  j)er- 
mit  us  at  the  later  period  to  buy  4  cwts.  cloves  and  no  cocoa,  that 
is,  this  time  a  lariicr  ([uantity  than  at  first,  and  one  just  doubles, 
as  the  other  was  half.  Had  he  done  so,  the  indication  of  depre- 
ciation would  have  been  no  stronger  than  that  of  appreciation. 
And  he  probably  failed  to  see  that  under  these  conditions  we 
could  at  the  first  period  purchase  with  (36.6G|  thalers  f  cwt. 
cocoa  and  with  133.33|^  thalers  IJ  cwts.  cloves,  or  2  cwts.  with 
200  thalers  ;  and  that  at  the  second  ]ieriod  we  could  purchase 
with  1  •■).■).•">.■>  1^  thalers  |  cwt.  cocoa  and  with  ()().66§  thalers  IJ 
cwts.  cloves,  that  is,  exactly  the  same  quantities  of  cocoa  and 
cloves,  amounting  to  2  cwts.,  v/ith  the  same  total  sum  of  money, 
200  thalers.  Had  he  noticed  this,  he  would  have  seen  that,  so 
far  as  either  he  or  Jev(ms  had  yet  carried  their  investigations, 
the  indicati<m  of  constancy  is  as  strong  as  the  indication  of  de- 
preciation, his  argument  offering  nothing  distinctive  in  proof  of 
the  indication  of  depreciation  made  by  the  arithmetic  average 
over  against  the  indication  of  constancy  made  by  the  geo- 
metric. 

And  when  Jevons  in  reply  suggested  the  harmonic  average 
(in  this  case  ..SO,  indicating  a  fall  of  prices  l)y  20  per  cent.)  on 
the  ground  that  it  marks  the  change  in  the  total  (piantity  of 
conunodities  the  same  sums  of  money  will  ])urchase  at  tlic  two 
periods  (liere,  for  100  thalers  spent  on  each  article,  1  cwt.  cocoa 
and  1  cwt.  (-loves  at  the  first  period,  and  at  the  second  -J  cwt. 
cocoa  and  2  cwts.  cloves,  the  arithmetic  average  l)eing  |  (J  -f-  '2) 
=  1.2"),  indi(;ating  appreciation  of  money  corres))ouding  to  the 
above  fall  ol'  prices),  he  probably  failed  to  sec  that  we  could  at 
the  first  [)eriod  |)urchase  with  13.']. .'>o^  thalers  1^  cwts.  cocoa- 
and  with  ()().()(! H  thalers  H  cwt.  cloves,  or  2  cwts.  with  200 
thalers  ;  and  that  at  tlic  second  |)eriod  we  could  purchase  with 
\:V^^^:^^  thalers  |  cwt.  cocoa  and  with  ()().()(v|  thalers  1^  cwts. 
cloves,  that  is,  with  exactly  the  same  expenditure  of  200  thalers 
the  same  total   (piantity  of  2  cwts., — wherefore  even  his  argu- 


MATHEMATICAL    RELATIONS    I(i.N<)IM;i)  269 

nient  for  the  harmonic  average,  so  far  as  he  worked  it  (Hit,  would 
indicate  constancy  as  readily  as  a  fall  of  prices. 

§  4.  AVe  have,  even,  as  yet  by  no  means  exhausted  the  possi- 
bilities Avhich  render  such  arguments  ridiculous.  The  following 
propositions  h(ild  : — The  price  of  at  least  one  class  rising  and  the 
price  of  at  least  one  class  falling,  no  matter  how  large  or  small 
these  variations  he,  it  is  possible  hij  spending  constant  sums  of 
money  on  the  different  classes,  tvith  the  same  toted  sum  to  purchase 
at  both  periods  tJie  same  total  quantity  of  goods  ;  and  under  the 
same  conditions,  it  is  possible  at  both  pteriods  to  purchase  constant 
quantities  of  the  different  classes,  consequently  the  same  total  quan- 
tity of  goods,  u-itli  the  same  total  sum  of  money,  differently  sjtcnt 
at  the  two  periods.  In  the  first  of  these  cases  the  equality  in  the 
total  quantities  of  goods  at  both  the  periods  depends,  given  the 
price  variations,  upon  two  other  factors.  It  depends  both  on 
the  sizes  of  the  mass-units  between  the  numbers  of  which  the 
compensation  by  arithmetically  equal  (piantities  is  desired,  and 
on  the  proportions  between  the  special  sums  devoted  at  both 
periods  to  purchasing  the  diflPerent  classes.  In  the  second  case 
the  equality  in  the  total  sums  of  money  at  both  periods  depends 
upon  only  one  factor  beside  the  price  variations.  This  one  other 
factor  is  the  proportion  between  the  special  quantities  of  the  dif- 
ferent classes  purchased  at  the  two  periods  ;  but  as  these  (piantities 
are  affected,  in  their  numerical  expressions,  by  the  sizes  of  the 
mass-units  used,  these  also  play  a  part,  though  a  subordinate 
one,  as  we  shall  see.  Now  this  factor,  so  far  as  it  is  a  factor, 
of  the  sizes  of  the  mass-units  used,  has  generally  been  decided 
at  the  outset  in  the  same  way  by  the  harmonic  and  by  thf^  arith- 
metic averagists.  They  both  employ  mass-units  that  are  equiv- 
alent at  the  first  period  (being  then  priced  at  1.00  or  at  100 
money -units) — as  we  have  just  seen  done  by  Jevons  and  Las- 
peyres.  These  massnmits  being  settled  upon,  the  first  factor  in 
the  first  case,  and  so  fiir  as  it  affects  the  second,  is  disposed  of, 
and  now  the  equality  in  question  depends  upon  the  other  fiictor 
in  both  cases.  In  the  first  case,  if  an  article  rises  much  in  price, 
so  that  the  deficiency  in  the  quantity  of  it  purchasable  at  the 
second  period,  compared  with  the  quantity  purchasable  at  the 


270  Ai.Mii'MKN'rs   Foi;    riii-:  oiiiki;   ankiiacks 

firt^t,  !;>  large,  and  if"  another  falls  slightly  in  price,  so  that  the 
gaiii  here,  quantity  I'oi-  (|uantity,  is  small,  we  only  have  to  ex- 
tend the  qnantity  of  this  class  purchased  at  the  first  |)erio(l  until 
its  gain  at  the  second  j)eriod  e(iuals  the  dcHcieucy  of  the  other. 
And  similarly  in  the  second  case.  The  number  of  classes  does 
not  affect  the  matter.  All  l)ut  one  may  rise  nuich  and  that  one 
fall  but  slightly  :  still  it  is  possible  for  the  comj)ensation  by 
arithmetic  equality  to  take  place. 

\^nien  we  are  dealing  with  only  two  oppositely  varying  classes 
it  is  easy  to  get  fornudje,  which  may  be  of  service.  To  begin 
with  the  first  case  : — Let  the  sum  of  money  devoted  at  both 
periods  to  purchasing  [A]  be  represented  by  a,  and  that  de- 
voted at  both  periods  to  ])iirchasing  [B]  be  represented  by  b  ; 
let  us  for  the  present  adopt  the  usual  course  and  tak(»  for  our 
mass-iniit  for  each  class  that  mass  of  it  which  can  be  purchased 
at  the  first  period  for  one  money-unit,  and  let  «./  and  fij  be  the 
prices  at  the  second  period  of  these  mass-units  of  [A]  and  [B] 
respective ely.      We    therefore   purchase  with    a    money-units  a 

a 

mass-units  of  [A]  at  the  first  period  and  at  the  second  ,  mass- 
units,  and  with  b  money-units  we  purchase  b  mass-imits  of  [B] 

b 

at  the  first  period  and  at  the  second  ,  ,  mass-units.       Assuminu' 

.  ^^ 

that    the  smns   of  these  inass-(|uantities  are   the   same   at   both 

periods,  \vc  have 

a         b 

a  +  b=      ,+   :.,, 

which  irives  us 

a,V«-l) 
<(l-/5/)' 

N(tw  let  us  take  a  as  a  unit  sum  ;   then 

which  means  that  f()r  every  sum  of"  money  spent  on  [A],  the 
rising  article,  we  must  spend  a  so  many  times  larger  or  smaller 
sum   on   [B],  the  falling  article.      Or  if  we  represent  the  total 


iMATHKMATlCAL    RELATIONS    IGXOliED  271 

sum  of  money  to  be  spent  hv  S,  and  the  .sum  to  Ix-  spent  on 
[A]  hv  .s'a,  we  have 


wlienee 


_.s<(l-j9/). 


and  representing-  the  sum  to  he  spent  on  [B]  hy  •%,  wv  easily 
ohtain 


'b  — 


a./  -  /9/ 


For  example,  suppose  A  rises  in  price  from  l.UO  to  l.!»!»  and 
B  falls  from  1.00  to  .99.  Here  all  the  employers  of  the  above 
arguments  would  at  once  conclude  that  the  level  of  prices  has 
risen  and  money  depreciated — the  harmonic  averagist  amongst 
them.  Yet  it  is  possible  with  the  same  sums  of  money  to  get 
the  same  total  (piantitv  of  the  goods  at  both  periods.      F(M"  iiere 

.99(1.99-1)       .9801 

b'  =        ^ -  = =  49.-J.)12.). 

1.99(1 -.99)       .0199 

Thus  if  we  spend  100  money-units  on  [A]  at  eaeii  period  we 
get  at  the  first  period  100  A  and  at  the  second  50.2512  A,  and 
if  we  spend  4925.1 2|  money-units  on  [B]  we  get  at  the  first 
period  this  quantity  of  B  and  at  the  second  4974. 87."^  ;  and 
the  total  quantity  of  [A]  and  [B]  bought  at  the  first  period  is 
5025.121,  and  the  total  quantity  of  [A]  and  [B]  bought  at 
the  second  is  the  same. 

It  is  plain  that  if  at  both  periods  we  devote  all  our  money  to 
purchasing  the  article  which  has  risen  in  price,  we  get  the  ex- 
treme diminution  in  the  total  (piantity  piu'chasable  at  the  second 
period  compared  with  the  total  quantity  purchasable  at  the  first ; 
and  if  at  both  periods  we  devote  all  our  money  to  purchasing  the 
article  \vhich  has  fallen  in  price,  we  get  the  extreme  augmenta- 
tion in  the  total  quantity  purchasable  at  the  second  period  et)m- 
pared  with  the  total  quantity  j)urchasable  at  the  first,  between 
these  extremes  it  is  evident,  bv  the  law  of  coutinuitv,  that  there 


272  AH(;iMENTs  Foi;    iiii;  oiiiEi;  AVi:i{A(ii;s 

must  be  some  {listribution  of"  mir  spciulinu-s  which  will  g-jve 
neither  diminution  nor  ;uiu'ni<'iitatii>n  in  tlic  Idtal  (|n:intities. 
This  is  the  (listrihntion  indicated  l)y  oiii' t'ni'iiuila.  it  i- evident, 
fnrther,  that  it"  we  spend  more  on  the  artiele  rising  in  [)rice  than 
the  proportion  iudieated  by  the  fornuda,  we  get  a  smaHer  total 
cpiantity  at  the  second  period,  and  more  and  more  smaller  as  we 
depart  from  this  proportion,  np  to  tlu'  limit  when  we  spend  all 
on  this  artiele.  And  reversely  il'  we  spend  more  on  the  article 
falling-  in  ])rice  than  the  proportion  indicated,  Ave  get  a  larger  total 
quantitv  at  the  second  period,  increasing  np  to  the  limit  when 
we  spend  all  on  this  artiele.  Thus  between  the  two  limits  there 
is  an  infinity  of  total  quantities  that  may  be  purchased  with 
exactl}'  the  same  sums  of  money  at  both  periods. 

§  5.  In  the  second  case,  let  the  mass-unit  used  for  each  class 
be  the  same  as  in  the  preceding  ease,  that  is,  the  mass  whose 
price  at  the  "first  period  is  one  money-unit,  and  again  let  the 
prices  of  these  mass-units  of  [A]  and  [B]  at  the  second  period 
be  «/  and  ^i.,'  res))ectively  ;  but  let  the  number  of  the  mass- 
units  of  [A]  to  be  purchased  at  both  j^eriods  be  represented  by 
x' ,  and  the  number  of  the  mass-units  of  [B]  to  be  purchased  at 
both  periods  be  represented  by  //'.  Then  the  sums  of  money 
needed  to  })urchase  x'  A  is  at  the  first  period  x'  money-units 
and  at  the  second  .r'a^'  money-units;  and  th(>  sum  of  money 
needed  to  purchase  y'  B  is  at  the  first  period  //'  monty-units  and 
at  the  second  ij' fi.'  money-units.  Assuming  that  the  total  sums 
of  these  sums  are  the  same  at  both  ])erio(ls,  we  have 


which  irives  us 


x'  +  ii'  =  x'a:  + 1,'^:, 

^=  1-.V  • 

And  now  again,  if  we  take    x'  as  a  (piantity-nnit,  we  have 

"-.!  -  1 

w^hich  means  that    foi-  every  ([uantity  of   [A]  we  purchase  we 
must  pnrchase  a   so   many  times   larger  or  smaller   (piantity  of 


MATIIK.AIATICAJ.    IJELATIO.NS    IGNOREJ)  273 

[B]  (or  must  spend  so  mucli  more  or  less  money  on  [B]  than 
on  [A]  at  tlu'  first  })eri(»d).  Or  if  we  represent  the  total  quan- 
tity of  both  classes  l)y  il,  and  the  quantity  to  be  purchased  of 
[A]  by  ry  ,  we  have 

whence 


7.c  = 


«/  -  iV 


and  rej)resenting  the  quantity  to  be  purchased  of  [B]  by  q^,  we 
easily  obtain 

_  QK  - 1) 

^^      «/  -  /?/  • 

As  these  expressions  give  the  quantities  of  equivalents  of  the 
first  period,  they  also  represent  the  sums  that  must  be  spent  at 
the  first  period. 

Thus  in  the  above  numerical  example,  in  which  A  is  sup- 
posed to  rise  in  price  from  1.00  to  1.99  and  B  to  fall  from  1.00 
to  .99,  and  in  which  the  arithmetic  averagist  would  probably 
see  only  a  rise  in  the  general  level  of  prices,  it  is  possible  to 
get  the  same  quantities  of  [A]  and  [B]  at  both  periods.  For 
here 

,,      1.99-1       .99 
y    -  1  _  .99  ~  .01  -  '^^  ' 

that  is,  if  we  purchase  100  A  at  each  of  the  periods,  we  spend 
for  it  100  money  units  at  the  first  period  and  199  at  the  sec- 
ond ;  and  if  we  purchase  9900  B  at  each  of  the  periods,  we 
spend  for  it  9900  money-units  at  the  first  period  and  9801  at 
the  second  ;  and  so  at  the  first  period  we  spend  for  the  quanti- 
ties of  [A]  and  [B]  together  the  total  sum  of  100  -|-  9900 
=  10,000,  and  at  the  second  the  total  sum  of  199  +  9801 
=  10,000,  or  in  other  words,  for  10,000  money -units  we  can  at 
each  period  jiurchase  100  A  and  9900  B. 

Here,  too,  it  is  plain  that  if  at  both  ]>eriods  we  ]>urchase  only 
the  article  which  rises  in  price,  we  have  the  extreme  augmenta- 
tion in   the  total   sum  of  money  needed   to  })urchase  the  same 
18 


274  ARCIMENTS    FOR    TIIK    oTIIKK    AVK1;A(  i  KS 

tjuantity  at  tlie  second  j)CTi()d  compared  with  the  total  sum 
needed  to  ])urc]iase  it  at  tlic  first ;  and  it"  at  hotli  |K'riods  wc 
purchase  only  the  article  whicli  falls  in  price,  we  have  the  ex- 
treme dimiuutiou  in  the  total  sum  of  money  needed  to  purchase 
the  same  quantity  at  the  second  period  comj^ared  with  the  total 
sum  needtxl  to  purchase  it  at  the  first.  Between  these  extremes, 
afiain,  it  is  evident,  by  the  law  of  continuity,  that  there  must  be 
some  distribution  of  our  purchases  which  will  re(iuire  neitlier 
augmentation  nor  diminution  in  the  total  sum  of  money  needed 
at  the  two  periods  to  purchase  the  same  quantities.  This  is  the 
distribution  indicated  by  our  formula.  Ajrniu  it  is  evident  that 
the  further  we  de])art  from  this  distribution  of  our  purchases  by 
purchasing  at  both  periods  more  of  the  article  rising  in  price, 
the  greatin-  will  be  the  total  sum  of  money  re(piired  at  the  sec- 
ond period  compared  w^ith  the  first,  up  to  the  limit  when  we 
purchase  only  this  article  ;  and  reversely,  by  j)inx'hasing  at  both 
])eriods  more  of  the  article  fallmg  in  price  the  smaller  will  be 
the  total  sum  needed  at  the  second  period  com])ared  with  the 
first,  up  to  the  limit  Avhen  we  purchase  only  this  article.  Thus 
between  the  two  limits  there  is  an  infinity  of  total  sums  that 
maybe  needed  to  purchase  exactly  the  same  <|iiantities  of  articles 
at  both  ])eriods. 

Kvidently  no  arguments  have  validity  that  rest  merely  on 
some  possibilities — either  on  the  potentiality  of  certain  sums  of 
money  to  purchase  (piantities  of  goods,  or  on  tli(>  potentiality  of 
certain  quantities  of  goods  to  connnand  sums  of  money. 

TIT. 

§  1.  Of  the  arguments  as  hitherto  em|)loyed  the  defectiveness 
may  be  mired.  It  consists  priniarilv  in  the  neglect  of"  weight- 
ing. Therefore  we  nuist  (ii'st  of  all  introduce  into  them  con- 
sideration of  weighting. 

Disregard  of  weighting  we  have  seen  to  be  the  fault  with  the 
objections  that  have  been  urged  against  the  em|)loyment  of  any 
average  as  an  indicator  of  constnnc\-  or  \:irintion  ol"  exchange- 
value.  Some  persons  haxc  |)layc<l  with  the  same  price  varia- 
tions, ditferently  representing  them,  and  really  representing  them 


COJjrvEtTlOX    OK    THEIR    DEFECTS  275 

SO  as  to  express  fliiferent  weit>'litin<!:s  ;  and  o-ettiiiu'  ditierent  re- 
sults, have  (leiiouncccl  the  whole  subject  as  intractable.  Similar 
disregard  of  weighting  has  been  the  besetting  sin  in  all  the  argu- 
ments for  each  of  the  averages,  and  the  cause  why  none  has  been 
convincing. 

Thus  in  the  controversy  above  reviewed  neither  Jevons  nor 
Laspeyres  sought  for  any  more  data  than  the  mere  variations  of 
prices.  Their  dispute  was  very  much  as  if  they  posited  that 
some  men  are  six  feet  tall  and  others  are  five  feet  tall,  and 
quarreled  t>ver  the  average  tallness.  Factors  necessary  for  the 
solution  of  the  problem  ^vere  absent ;  yet  they  blissfully  went  on 
with  the  attempt  to  solve  it.  They  each  relied  on  various  ])()ssi- 
bilities  in  the  purchases,  instead  of  requiring  to  have  given,  as 
real  or  as  suppositional  data,  what  were  the  actual  purchases. 
Or  if  they  did  think  they  were  agreeing  upon  the  use  of  even 
weighting,  they  each  conceived  of  this  diiferently.  Laspeyres 
supjiosed  e(|ual  mass-quantities  to  be  purchased  of  both  articles, 
and  constantly  so  at  both  periods  (and  also  equal  thaler's  worths, 
but  only  at  the  first  period).  Jevons,  when  he  suggested  the 
harmonic  average,  supposed  equal  sums  of  money  to  be  spent 
on  both  classes,  and  constantly  so  at  both  periods  ;  but  when  he 
ad\'ocated  the  geometric  average,  he  gave  no  hint  \vhat  concep- 
tion he  had  of  even  weighting.  Such  argumentation  disparages 
the  whole  subject  of  exchange- value  mensuration,  and  strengthens 
the  hands  of  its  opponents. 

In  general,  writers  on  the  subject  have  made  arguments  for 
the  different  averages  without  regard  to  weighting,  and  they  have 
made  arguments — if  arguments  they  deserve  to  be  called — for 
several  kinds  of  weighting  without  regard  to  the  averages.  They 
have  never  combined  them.  Up  to  this  point  separate  argu- 
mentation has  been  employed  in  this  work  also.  We  now  need 
to  make  the  combination. 

§  2.  It  is  plain  that  in  the  argument  made  by  the  harmonic 
averagist  the  weighting  is  according  to  the  constant  sums  of 
money  that  are  supposed  to  be  devoted  at  both  periods  to  purchas- 
ing variable  quantities  in  the  difFerent  classes. 

This  being  so,  certain   peculiarities  arise,  due   to  the  other 


276  ARGUMEN'1><    FOR    THE    OTIIKl!    A  Vi:i;A(i];s 

factor  in  this  ease,  nauiely  tiic  sizes  of  the  mass-units  used,  whose 
numbers  are  the  mass-quantities  between  which  compensation  is 
desired  by  their  e(|iiality.  These  peculiarities  may  here  be 
briefly  indicated. 

Let  .i\  and  x.„  y,  and  i/.„ represent  the  numbers  of  mass- 
units,  wliatever  these  be,  of  the  classes  [A],  [B], ,  purchased 

at  the  first  and  at  the  second  periods  respectively  with  the  con- 
stant sums  a,  b, The  only  conception  Ave  as  yet  have  of 

the  mass-quantities  arc  represented  by  these  symbols  x^,  x^,  y^,  y^, 
And  so  the  argument  from  compensation  by  equal  mass- 
quantities  for  constancy  in  the  exchange-value  of  money,  or  for 
variation  in  this  exchange-value  by  the  variation  in  the  total 
mass-quantities,  calls  for  formulation  as  follows, 

Mq2    ^-^'2  +  ^2  +   .-.X 

M,,         x^+y^^...-'  ^') 

whence,  })y  inversion. 

But  unless  we  specified  that  we  had  already  selected  the  proper 
mass-units  for  this  purpose,  that  is,  if  we  only  used  the  ordinary 
mass-units  such  as  may  be  variously  used  by  merchants,  it  is 
evident  that  in  such  formuhe  the  result  would  be  variable  ac- 
cording to  the  sizes  of  the  mass-units  we  happened  to  use,'  and 
we  should  be  committing  the  same  sort  of  absurdity  as  committed, 
from  the  other  side,  by  Dutot."  We  must  add  a  restriction  to 
these  formuhe  by  including  in  them  the  method  of  selecting  the 

mass-units.      Let  «,  and  (/..„  fi^  and  ^9.,, be  the  prices  at  the  first 

and  at  the  second  ])('riods  respectively  of  the  mass-units  df  [A], 

a 
of  [B],  which    we   do   liajipcn    to   use.      Th(Mi   ,r,  =   —  and 

abb 

.T„  =  —  ,  w,  =  —  and  //,=  —,  and  so  on.       If  now  \\v  ('(inverted 

«2  P\  Pz 

the  last  fornuila  into  this, 

1  8ee  C'liapt.  V.  Sect.  III.  i^.'). 

2  Sec  Chii])!.  v.  Sci-t.  VI.  iJ.i. 


CORRECTION    OF    THEIR    DEFECTS  277 


p. 

a       b 

p. 

a       b 

-  +  -0  + 

(3) 


we  should  still  have  no  single  determinate  result.'*  The  prin- 
ciple of  selecting  the  mass-units  is  still  missing.  A  simple  sug- 
gestion is  that  we  should  select  mass-units  that  are  equivalent. 
But  then  the  questions  arise,  Equivalent  at  one  of  the  periods 
only?  and  at  which  ?  or  over  both  the  periods  together?  Now 
without  consideration,  without  offering  a  reason,  without  argu- 
ment, people  have  agreed  upon  the  convenient  practice  of  em- 
ploying mass-units  that  are  equivalent  at  the  first  period.  This 
being  done,  and  in  the  still  more  convenient  form  of  employing 
mass-units  that  are  ecpiivalent  to  the  money-unit  at  that  period, 
so  that  a^'  =  fi^'  = =  1,  the  last  formula  reduces  to 

P,  _    a  +  b  + 

P,  ~    a       b 

or  by  using  n"  to  represent  a  -f  b  -|-  , 


P., 


Pi 


-^(  a     ,  -hb^^-h ) 

n"  \     «/  it^  ) 

and,  by  restoring  the  price  variations  to  their  original  forms, 
P„  1 


(4) 


(5) 


„  ^  a  -^  +  b  ^  + 


These  two  are  formulae  (15,  i)  and  (16,  i)  given  in  Cha})ter  V. 
Section  V.  §  1,  as  the  formulae  for  the  harmo)iic  average  of 
price  variations  with  weighting  according  to  a,  b, 

This  is  why  this  argument,  applied  in  this  way,  is  an  argu- 
ment for  the  liarmonic  average  of  price  variations.  But  there 
is  no  necessity  for  it  to  be  applied  in  this  way,  for  which  no 

3  Cf.  Chapt.  v.  Sect.  VI.  ^  2. 


278  ARor^fKXTs   Foi;  Tin:  (iriiKi;   avki:a<;ks 

reason  hiMi  been  oti'erotl,  so  as  to  Ixvoine  an  arunnnent  s})eeially 
for  the  harmonic  average.  Api)lie(l  to  otlier  niass-units,  it  may 
become  an  argument  for  other  averages. 

Thus  suppose  we  select  mass-units  that  are  equivalent  to  the 

money-unit   at  the  seccmd  ])erio(l.      Then  a.^'  =  fi^'  =  =  1, 

and  the  above  formula  (3)  rednees  instead  to 

a        b 

P^  _  <  +  ^/  + 

P,  -  a-^b  + ' 

or,  by  using  n"  to  represent  a  -f-  b  -|-  ,  and   restoring  the 

price  variations  to  their  simpler  forms, 

^.K^^''^ >      ^«) 

which  is  formula  (16, 2)  given  in  Chapter  V.  Section  Y.  ^  1,  as 
the  fornuda  for  the  aritkmetk-  average  of  ])riee  variations  with 

weigliting  according  to  a,  b, 

Or  again  we  might  equally  well,  for  all  we  have  as  yet  heard 
to  the  contrary,  and  perhaps  better,  select  mass-units  that  are 
equivalent  over  both  the  periods  com[)ared.  The  method  of 
getting  these  has  been  examined  in  Chapter  IV.  They  are 
such  that  the  geometric  means  of  their  prices  are  ecpial.  This 
suggests  some  relationship  with  the  geometric  mean  or  axx-rage 
of  price  variations.  Now  we  do  not  find  any  exact  connection 
here  with  the  geometric  (tveirn/c — that  is,  when  we  are  dealing 
with  several  classes  or  with  uneven  weighting.  But  when  we 
deal  with  oidy  two  classes  eveidy  weighted  (that  is,  in  this  case, 
equally  large  at  each  jieriod),  so  as  to  be  able  to  employ  the 
geometric  mean  of  the  price  variations,  the  connection  is  per- 
fect. For  now  we  have  a  =  b,  whereibre  we  may  represent 
them  each  as  1  ;  and  from  the  nature  of  the  mass-units  (because 
of  whose  peculiarity  we  may  distinguish  their  prices  and  mass- 
quantities   l)y  doubly  priming   them)  we   have  w/'^o"  =  /V/Vj 

,0  //  0  //  0  If  0  II 

whence  ^/,"  =  '  '   ',;    and  ^/,/' =  '  '  Vi^ .       Bv   substituting    the 

first  of  these  values  in  fornuda  {'A)  we  get 


COUltKCTlON    OF    TIlKli:     DKFKCTS  279 


1  <  +  /V 

1^    i^:W  "^  iV      /VW 


P    ~         1  1        ~  yV  +  <'       l^y"  ' 


(6) 


"•2  /'a 

and  1)V  siibstitutinti'  tlic  second, 


/V  '^2"/^2" 


1     1     .v'+j'/: 


(7) 


Hence  we  have,  together, 


P.,  _  '/./'       fi 


whence 

i>  ,,  //    7  To  // 


(8) 


which  we  should  also  have  obtamed  directly,  had  we  simul- 
taneously substituted  the  two  values/  (Here  in  parenthesis  we 
may  notice  that,  as  x/'a/'  =  x^'a.y  =  a  =  1  and  ///'/9/'  =  yj'fi.!' 

=  b  =  1 ,  we  have  «/'  =   _  j, ,  «/'  =        ,  fif'  =         ,  and  /9/'  = 

•'  1  ''2  "1 

y, ;  wherefore  by  substituting-  these  values  in  the  formulae  (()j, 
(7),  and  (8)  we  get 

P         II  "       X  "       X "  4-  II " 

Pi  -V  //2'''  ■^"2"   +   //2'"'' 

the  last  part  of  which  Ave  knew  already,  according  to  the  hy- 
pothesis.) Now  from  the  above  combination  of  formuhe  ((i)  and 
(7)  we  also  form 

/p, Y^ «/'  ,V' 

whence  we  draw 

P, 
P 


Jo  "  H  " 


*  Tliis  is  Dutot's  luethod,  which,  therefore,  is  rational  in  tiiis  one  special  ease. 


280  ARGUMENTS    FOR    TlIK    OTHER    AVERAGES 

Avhicli  is  tlu'  tonmila  for  tlie  (/coiiicfric  nicfoi  of  the  price  varui- 
tions  of  tlie  two  classes,  witli  even  \vei<>-htino-,  hcinn  like  the 
formula  (14,;!)  in  Chapter  V.  Section  IV.  §  2. 

Thus,  when  attention  is  paid  to  the  wei}i:htini)-,  this  aru'unient 
from  compensation  hv  e([ual  massHpiantities  turns  out  to  he  still 
an  argument  ecpiaily  well,  as  yet,  a])plicable  either  to  the  har- 
monic or  to  the  arithmetic  means  or  averages  or  to  the  geometric 
mean.  Therefore  in  order  to  decide  what  a\'erage  or  mean  it 
rcallv  favors, — or  if  it  favors  the  geometric  mean  in  the  case  of 
two  equally  important  classes,  what  is  the  method  it  favors  in 
the  case  of  inauv  variously  important  classes, — we  must  pay  at- 
tention not  only  to  the  weighting,  hut  also  to  the  mass-units  it 
is  proper  to  use.  We  must  search  for  a  reason  why  one  set  of 
mass-units  is  to  be  preferred,  conducting  an  investigation  which 
has  hitherto  been  neglected. 

§  8.  In  the  argument  made  by  the  arithmetic  averagist  the 
weights  cannot  be  according  to  the  constant  mass-quantities  that 
are  purchased  at  each  period  by  \'ariable  sums  of  money  (unless 
we  know  some  way  of  properly  selecting  the  mass-units  for 
this  pur[)()se),  nor  can  they  be  merely  according  to  the  sums  of 
money  spent  on  them,  because  these  are  different  at  the  two 
periods.  Yet  they  must  be  somehow  connected  with  these  sums 
of  money,  as  we  have  already  examined  in  Chapter  LV.  The 
formula  which  represents  the  conditions  to  whic^h  this  arguments 
is  apj)lied  and  expresses  its  treatment  of  them  is  easily  seen  to 
be  either  this, 

^==^=+^+ (11) 

P,      a,  +b,  -f ^     ^ 

or,  more  definitely, 

P,      ■>•«,  +  .'/i',  + ^     ' 

in  which  x,  y, represent  the  numbers  of  times  the  ordinary 

commercial    mass-units    of    [A],     [J>], ,   whose    prices   are 

«,  and  a.,,  fi^  and  /9.„ ,  at  the  lirst  and  second  |)ei"iods  re- 
spectively. Now  this  last  is  the  formula  for  Scrope's  method, 
discussed  in  Chapter  V.  Section  Y  I.  §  4,  and  there  shown  to 
represent,  or  to  be  identical  with,  the  arithmetic  mean  or  average 


COlMiJ-X'TION    OF     I'llKIU    DKI'KCTS  281 

of  the  price  variations  with  \vei»>litiii<;'  aeeordiiifr  to  the  sums  of 
money  spent  on  the  constant  mass-quantities  in  the  different 
classes  at  the  first  period,  I)ut  that  analysis  was  not  complete. 
We  shall  later  find  that  this  f'ornuda  (Hpially  well  represents  the 
h(frmo)ii('  averai;e  or  mean  of  the  })ri(!e  variations  with  wei»i:hting 
according  to  the  sums  spent  at  the  .second  period,  and  again,  in 
some  cases  (namely  when  we  arc  dealing  with  only  two  equally 
important  classes),  the  geometric  mean  with  (even)  weighting  ac- 
cording to  the  geometric  mean  of  the  sums  at  both  the  periods. 

Thus,  again  in  this  case,  with  attention  paid  to  the  weighting, 
the  argument  is  still  applicable  as  \v(^ll  either  to  the  arithmetic 
or  to  the  harmonic  means  or  averages  or  to  the  geometric  mean. 
Here,  however,  we  are  no  hunger  bothered  by  the  question  of 
the  mass-units  to  be  used  ;  and  yet  we  shall  find  that  the  selec- 
tion of  the  mass-units  has  had  something  to  do  with  producing 
the  appearance  of  this  argument  being  more  specially  in  favor 
of  the  arithmetic  average.  Here,  too,  we  shall  see  that  the  ques- 
tion is  not  between  different  kinds  of  averages  vielding-  different 
results,  but  between  three  different  interpretations  of,  or  three 
different  ways  of  paralleling,  one  result  yielded  by  one  method. 

§  4.  Thus  in  general  we  have,  not  an  argument  for  the  har- 
monic average  and  an  argument  for  the  arithmetic  average  of 
price  variations,  as  their  emjiloyers  have  hitherto  conceived  them 
to  be,  but  t^vo  arguments  for  either  of  the  three  averages,  or 
means,  applicable  the  one  to  one  state  of  things  and  the  other 
to  another, — between  which  arguments,  therefore,  Avhen  rightly 
confined  each  to  its  owni  field,  there  cannot  even  be  contradic- 
tion, or  antagonism. 

The  fields  to  which  these  arguments  are  by  their  own  natures 
confined  remind  us  of  the  first  two  divisions  in  the  question  of 
weighting  wdiich  w^e  discussed  in  Chapter  IV.  Section  V.  But 
those  two  divisions  in  the  question  of  weighting  we  recognized 
to  be  incomplete.  Similarly  these  two  arguments,  correspond- 
ing to  those  two  divisions,  are  incomplete.  The  argument  made 
by  the  harmonic  averagist  supposes  that  we  spend  the  same  sums 
of  money  on  every  class  at  both  periods  in  spite  of  the  variations 
in  their  prices,  which  w^e  rarely,   if  ever,  do.      The  argument 


282  ARdUMEXTS    FOR    THE    OTHER    AVERAGES 

made  by  the  arithmetie  averaj>;ist  supposes  that  we  huy  the  same 
quantities  of  every  class  at  both  periods  in  spite  of  the  varia- 
tions in  (heir  jiriees,  which  we  rarely,  if  ever,  <\o.  \<  a  riuiu-h 
proposition,  we — aconiniunity — «ienerally  spend  more  on  articles 
that  have  risen  in  price  and  get  less  of  them,  and  spend  less  on 
articles  that  have  fallen  in  price  and  jj^et  more  of  them.  At  all 
events,  we  almost  always  spend  our  money,  and  buy  iioods,  in 
different  ])roportions  at  any  two  periods.  Thus  the  more  usual 
state  of  thino-s  is  neglected  bv  the  arp-uments  as  made  bv  the 
harmonic  and  the  arithmetic  averagists,  except  in  a  possible  sup- 
plement to  each  argument  by  \vhich  the  moi'e  complex  state  is 
reduced  by  curtailment  to  the  one  or  to  the  other  of  the  states 
required  by  these  arguments.  Then,  as  already  remarked,  the 
arguments  become  antagonistic  ;  but,  so  used,  neither  will  have 
much  claim  for  our  respect. 

Now  the  third — the  more  usual,  the  more  complex — state  of 
things  is  really  made  up  of  each  of  the  other  two,  in  this  way  : — 
In  the  first  state  of  things  above  described  the  particuhir  mass- 
quantities  are  different  at  the  different  periods,  and  in  the  second 
state  of  things  the  particular  sums  of  money  are  different  at  the 
different  periods  ;  and  this  double  difference  is  precisely  what  ex- 
ists in  the  more  complex  state. 

Hence,  even  though  the  two  states  of  things  i'c([uircd  in  the 
two  arguments  are  not  likely  ever  to  hap})en  in  reality,  it  is  well 
for  us  to  examine  these  arguments  thoroughly,  just  as  if  their 
states  were  likely  to  occur ;  because  after  reaching  the  right 
method  for  eacii  of  these  states,  we  shall  be  in  a  ])osition  to  com- 
bine them,  and  so  form  the  right  univci'sal  method,  applicable 
to  the  complex  states  which  generally  exist. 

Thus  our  future  work  is  ma])ped  out  for  us.  We  must  ex- 
amine; the  two  arguments  separately,  each  apphed  to  its  own 
state  of"  things  ;  and  then  we  shall  seek  what  can  be  luiited  of 
the  two  for  the  com[)lex  state.  The  thi-ee  divisions  of  weight- 
ing, which  we  |)reviously  (lis|)ose<l  of,  so  fin"  as  then  possibh',  in 
one  Section  of"  one  ( 'ha|)ter,  now  become  dixisions  in  the  (piestion 
of  the  av(!rages,  will  occupy  the  next  three  ( 'hapters. 


CHAPTER   X. 

THE   METHOD   FOR  CONSTANT  SUMS   OF   MONEY. 


§  1.  Having-  examined  in  a  general  way  the  argument  from 
compensation  by  equal  mass-quantities,  we  must  now  examine  it 
in  detail,  applying  it  to  many  particular  examples,  for  the  pur- 
pose of  discovering  in  them  a  principle  that  will  give  us  a  clear 
indication  of  the  true  average  or  mean  or  method  for  measurino^ 
the  constancy  or  variation  in  the  exchange-value  of  money  in 
the  cases  when  constant  sums  of  money  are  spent  on  every  class 
at  both  the  periods  compared.^  Also  we  may  look  for  some 
crucial  instances,  serviceable  as  tests,  that  shall  render  our  con- 
clusions demonstrative.  We  must  survey  first  of  all  the  differ- 
ent ways  in  which  the  argument  is  applicable  to  the  different 
averages,  or  means,  in  order  that  we  may  later  be  in  a  position 
to  judge  in  which  application  the  argument  is  valid. 

By  compensation  by  equal  mass-quantities  it  cannot  be  meant 
to  judge  things  by  actual  masses,  counted  in'  any  of  the  usual 
weights  or  measures  ;  but  the  idea  is  to  compare  things  by  pro- 
portions of  masses.  If,  for  exam])le.  A,  purchasable  at  the  first 
period  with  one  money-unit,  be  a  quarter  of  barley,  and  B,  like- 
wise tlien  purchasable  with  a  money-unit,  be  a  bushel  of  wheat, 
a  loss  of  one  third  on  [A]  at  the  second  period  is  a  loss  of  2f 
bushels  of  barley,  while  a  gain  on  [B]  by  a  third  is  a  gain  of 

^  All  that  IS  said  in  this  Chapter  may  he  extended  also  to  cases  in  which  at 
both  periods  there  are  expended  on  all  the  classes  sums  in  the  same  proportion, 
whether  at  the  second  period  they  be  all  smaller  or  all  larger  than  at  the  first, 
provided  all  the  reasoning  (and  all  the  formuhe  later  to  be  described,  except  two) 
be  applied  only  to  the  sums  of  the  one  or  of  the  other  period — or,  theoretically 
better,  only  to  what  is  common  to  both  periods.  For  the  sums  in  excess,  all  being 
in  the  same  proportion,  have  no  influence  to  alter  tlie  exchange-value  of  money 
already  determined. 

283 


284  TIIK    METHOD    FOR    COXSTANT    S['MS 

only  J  huslu'l  of  ^\■lR■at.  \\'itli  two  nioney-uiiits  evenly  distrih- 
uted  we  eonkl  pureliase  at  the  first  period  1  A  +  1  B,  or  9 
bnshels  of  grain,  and  at  the  second  f  A  -|-  |^  R,  or  G^bnshels  of 
grain — a  loss  in  bnlk.  Or  if  A  be  one  ponnd  of  copper  and  B 
ten  pounds  of  iron,  a  loss  of  J  on  [A]  is  a  loss  of  ^  pounds  of 
eoi)per,  and  a  gain  of  jV  on  [B]  is  a  gain  of  ^^^  pounds  of  iron, 
so  that  from  })urchasing  1 1  pounds  of  metal  our  two  money- 
imits  evenly  distributed  will  come  to  purchase  14  pounds  of 
metal — this  time  a  gain  in  weight.  Thus  even  the  arithmetic 
ecjuality  in  tlu>  compensation  offered  by  the  harmonic  system 
disaj)j)ears  as  a  compensation  by  (|uantitics  of  masses  literally 
talvcn — or  rather,  taken  at  haphazard.  The  equality  is  system- 
atically obtained  only  by  treating  the  masses  of  [A]  and  [B], 
whatever  they  be,  })urchasable  with  the  same  sum  of  money  at 
some  period  or  periods,  as  equal  because  they  are  equivalent. 
That  is,  these  masses  are  conceived  as  equal,  not  as  weights  or 
capacities,  but  as  exchange-values.  For  then  the  equal  mass- 
quantities  are  coneeived  as  composed  of  equal  numbers  of  such 
equivalent  individuals.  A  loss  of  one  third  on  one  article  is 
supposed  to  be  compensated  by  a  gain  of  one  third  on  another, 
even  though  this  one  third  be  but  a  small  fraction  by  weight  or 
bulk  of  the  other  one  third,  or  even  though  it  far  exceed  the 
other  in  weight  or  bulk,  provided  it  be  as  much  more  or  less 
valuable  as  it  is  physically  smaller  or  larger,  so  as  to  be  equiva- 
lent to  the  other,  its  greater  precionsness  making  up  for  its  lack 
of  weight  or  bulk,  or  reversely, — at  some  ])eriod  or  periods. 
The  argument  is  really  in  its  proper  form  when  it  passes  from 
the  quantities  purchasable  to  the  powers  of  purchasing  them,  and 
claims  that  a  loss  of  a  third  in  the  })urehasing  power  of  money 
over  [A]  (or  its  exchange-vahu;  in  [A])  is  to  be  eonqx'nsated 
by  a  gain  of  a  third  in  the  purchasing  power  of  moiuy  over  [B] 
(or  its  exchange-vahie  in  [!)] ).  lint  as  we  measure  })articular 
exchange-values  or  purchasing  powers  of  money  by  the  quanti- 
ties of  the  things  a  constant  sum  will  purchase,  we  may  continue 
to  treat  of  the  conqx'usation  by  direct  reference  to  the  quantities. 
Xow  as  we  are  dealing  only  with  the  proportions  (»f  tlie  quan- 
tities, we  have   three  ways  of  conceiving  of  the   proportit)ns, 


SCHEMATA  2  .SO 

when  avowedly  dealing  with  eqnally  inijxtrtant  clas.se.<  : — either 
(1)  as  })r()})()rti()ns  of  variation //-o//;  an  eqnal  condition  at  the 
first  period,  or  (2)  as  proportions  of  variation  io  an  ecpial  cdndi- 
tion  at  the  second  period,  or  (3)  as  proportions  of  variation,  in 
any  two  classes,  in  the  one  from  a  certain  condition  to  another 
and  in  the  other  from  the  latter  condition  io  the  former.  The 
same  three  positions  are  obtained  by  snpposing  the  arithmetic 
compensation  by  eqnal  qnantities  to  be  in  eqnal  numbers 
of  mass-units  (ideally  constructed  for  the  purpose)  that  are 
equivalent  (1)  at  the^^rs^  period,  or  (2)  at  the  second  period,  or 
(3)  over  both  the  two  periods  together.  It  is  more  usual  to 
adopt  the  first  of  these  methods  of  measurement  and  this  is  the 
reason  why  this  argument  from  equal  mass-quantities  has  seemed 
to  be  an  argument  specially  favoring  the  harmonic  av^erage  of 
price  variations.  But  we  must  examine  all  three  of  these 
methods  in  turn. 

§  2.  Using  mass-units  that  are  cquira/ciif  at  the  first  period, 
we  may  construct  the  following  schemata  illustrative  of  the  con- 
ditions when  there  is  compensation  by  equal  numbers  of  such 
mass-units.  We  may  still  at  first  confine  our  attention  to  two 
classes  supposed  to  be  equally  large  in  some  respect.  In  the 
schemata  are  supposed  to  be  expended  at  each  period,  marked  I 
and  II,  certain  constant  sums,  which  are  stated  on  the  right- 
hand  side.  On  the  left  are  stated  the  mass-quantities,  that  is, 
the  numbers  of  these  mass-units  purchasable  with  these  sums  at 
their  prices,  also  stated,  at  each  period  ;  and  on  their  right,  or 
in  the  middle,  are  added  the  sums  or  totals  of  these  mass-quan- 
tities. Thus,  on  mass-units  equivalent  at  the  first  period  we 
have  compensation  by  e(|ual  mass-(piantities  when  the  ])rice 
variations  are  to  the  simple  harmonic  extremes,  as  follows  : 


1100    A  @  1.00     100    B@,  1.00  — 200 
II    ()GSA@1.50     133^  B@    .75  —  200 


100  for  [A]     100  for  [B], 
100  for  [A]     100  for  [B]; 


when  the  ])rice  variations  are  to  the  simple  arithmetic  extremes, 
as  follows  : 

I  100    A  (<i\  1.00     33i  B  0/,  1.00  — 133A  1  100  for  [A]     ;!3A  for  [B,], 
II    ()()§  A  («}  1.50     ()(3i  B  (<i     .50  — 133J  |  100  for  [A]     33A  for  [B]; 


286  THE  >[i:THor)  fou  coxstaxt  si'ms 

when  tlic  \)r\vv  varititioiis  are  to  the  simple  aeometrie  extremes, 
as  follows  : 

I  100    A  (>r  1.00       ()6s  B  (o,  1.00   —  16()f  I  100  for  [A]     f)6|  for  [B], 
II    663  A  fr,   1.50     100    BCn     .66s  — 166fi  I  lOOfor  [A]     66^  for  [B].« 

Here  we  have  the  peculiar  features  with  which  we  are  already 
familiar.  In  the  first  the  numbers  of  mass-units  are  ecjual  at 
the  first  j)eriod,  and  the  eompeusation  is  by  arithmetically  equal 
variations  in  them — over  equal  distances  away  from  this  equal 
condition.  In  the  second  the  numbers  of  mass-units  are  equal 
at  the  second  j)eriod,  and  the  compensation  is  by  harmonically 
equal  variations  in  them — over  e(jual  distances  uitino-  toward 
this  equal  ccmdition.  In  the  third  the  numbers  of  the  mass- 
units  alternate  and  change  places,  traversing  not  only  e(jual  dis- 
tances, but,  so  to  speak,  the  same  road,  so  that  in  them  the  com- 
pensation is  by  geometrically  equal  variations. 

The  universality  of  these  relations  existing  in  the  first  of 
these  particular  examjiles  has  been  demonstrated  near  the  end 
of  the  preceding  Chapter.  They  all  admit  of  demonstration  by 
means  of  one  of  the  formulse  discovered  earlier  in  the  same 
Chapter — in  §  4  of  Section  II.  In  the  formulation  there  made 
the  mass-units  were  su])])osed  to  be  equivalent  at  the  first  period, 
so  that  it  is  ap])lical)le  here.  I>ct  the  ])riee  of  A  always  be  sup- 
posed  to   rise  from  l.OO   to  o.,,' .      Then   if  the  ]U"ice  of  B  falls 

«' 

from   1.00  to  the   harmonic   extreme,  it  will  fall  to  ,     ,^  — -;   if 

-«.,    —  1 

to  the  arithmetic  extn-me,  it  will  fall  to  "2  —  a,,'  ;   if  to  the  geo- 
metric extreme,  it  will  fall  to      ,•    Supnlviuii' these  values  of  ;9./ 

'■'.,  II.. 

in  the  formida 

-  Here  in  nil  the  selieniatii  the  differences  in  the  numbers  of  A  and  of  B  are  the 
same,  and  the  totals  are  different.  Arrangement  can  be  made  so  that  the  totals 
would  lie  tlu-  same,  as  follows  for  the  second  and  third  schemata  : 

I    l.'.d  A  (w  1.(K)      50  B  («;  1.00    -200  I  ir)0  for  [A]     .")()  for  [B], 
II    100  A  0(  l..'>o     100  B  (a    .50     -200  |  150  for  [A]     50  for  [B]  ; 

I   120  A  (oj  1.00      SO  15  (a,  1.0.0   -200  I  120  for  [A]     80  for  [B], 
n     so  A  @  1.50     120  B  (o)    .66!i  -200  |  120  for  [A]     SO  for  [B]. 

But  here  the  differences  are  ilifferent.     The  arraiiKement  employed  in  the  text   is 
more  perspicuous. 


SCHEMATA  287 

~<(l-/5,')' 
we  find,  wlien  the  price  variatinns  are  to  the  harnionie  extremes, 

V=l; 
when  they  are  to  the  arithmetic  extremes, 

2  —  ry  ' 

b'  =  - -'    ■ 

f        y 

when  tliey  are  to  the  geometric  extremes. 

The  first  (»t"  these  expressions  means  that  when  the  price  varia- 
tions of  two  classes  are  to  the  opposite  harmonic  extremes,  in 
order  to  have  conn)ensation  liy  equal  numbers  of  mass-units 
equivalent  at  the  first  period  \ve  must  spend  our  money  evenly 
on  the  two  classes  at  l)()th  |)eriods  ;  wherefore  we  must  j)urchase 
at  the  first  jteriod  equal  numbers  of  their  mass-units,  and  at  the 

1  2«/— 1 

second  i)eriod      ,  A  and  — ^— , —  B,  which  are  arithmetic  extremes 
«./  «./ 

around  1,  since  half  their  sum  is  1.  The  second  means  that 
when  the  price  variations  are  to  the  opj)Osite  arithmetic  ex- 
tremes, in  order  to  have  compensation  by  equal  numbers  of  such 
mass-units  we  must  spend  at  both  periods  for  every  1  M  on  [A] 

2  -  «./ 

—    ,  -  M  on    TBI  ;  wherefore  the  numbers  of   the  mass-units 

purchased  at  the  first  period  are  in  these  pro})ortions,  and  at  the 

1  '/,/  1 

second  i)eriod  thev  are      ,  A  and  - — = — ;  =       ,  B,  that  is,  thev 
■  «.,  2  —  ^/.,  '/.., 

are  c(iual    numbers   at  this  i^eriod  ;    and    now! ,=      , — 

2  -  O.J 

^,  ",  which  shows  that  the  numbers  at  the  first   period   arc 

arithmetic  extremes  around  the  common  number  at  the  second. 
The  third  means  that  when  the  price  variations  arc   to  the  o])- 


288  THE  :METHor>  foe  constant  sums 

posite  <2:c()mc'tri('  extremes,  in  onlci"  to  have  eompeiisation  by 
e(ju:il  numbers  of  sueli  mass-units,  we  nnist  sj)en(l  at  both  periods, 

for  every  1  M  on   [A]      ,  ]\I  on   [B]  ;  wherefoi'e  tlie  nnml)ers  of 

the  mass-units  j)urchase(l  at  the  liist  |)erio(l  are  in  these  ])roj»or- 

1 

1  «./ 

tious,  and   at  th(>  second  tlicn-  ai'e      ,  A  and  ^^  =  1  B,  so  that 

'/.,  1 

these  numbers  alternate  over  the  two  periods.  ^  We  may  there- 
fore use  the  above  particuhir  examples,  and  the  rehitions  found 
in  them,  as  universally  illustrative. 

Now  if  on  the  price  variations  in  them  supposed  we  employ 
the  harmonic  average  on  the  first,  the  arithmetic  on  the  second, 
and  the  geometric  on  the  third,  all  with  even  weighting,  we  get 
in  every  instance  an  indication  of  constancy.  But  we  have  no 
right  to  use  even  weighting  in  every  case.  The  only  reason 
we  can  find  for  using  even  weighting  in  all  these  cases  is  that  in 
all  of  them  the  same  munbers  of  these  mass-units  are  purchased 
at  some  period  or  periods, — in  the  first  at  the  first  period,  in  the 
second  at  the  second,  in  the  third  at  each  of  the  ])eriods  alter- 
nately. But  of  course  there  is  nothing  to  reconnnend  such  a 
combination  of  weighting  and  of  averaging,  it  being  remembered 
that   these   mass-units   are  equivalent    oidy  at   the   first  j)eriod.' 

3  In  this  last  case  if  tlie  total  sum  to  be  spent  on  the  two  chisses  is  2.0(1,  we  find 

that  at  both  the  periods  we  must  spend    ~,   "  ,  M   on  FAl   and  — r^-—  ^I  on    [H], 

((■>   ^1  "2    ^1 

getting  these  numbers  of  A  and  B  at  the  first  jjeriod,  and  at  the  second  the  reverse. 
These  figures,  rather  curiously,  are  the  liarinonic  means,  the  first  l)etween  1  and 

fto',  the  second   i)etween    1   and",  (^ji./)-      Tliey  ai-e  also  arithmetic  extremes 

around  1.  (Thus  the  harmonic  means  between  unity  and  geometric  extremes 
around  unity  are  arithmetic  extremes  around  unity.) 

■*  Another  identity  in  the  results  deserves  notice.  In  all  three  cases  if  we  use 
tlie  harmonic  average  with  weighting  in  each  ease  according  to  the  numliers  of 
these  mass-units  at  the  first  period,  or  if  we  use  the  arithmetic  average  with 
weighting  in  each  case  according  to  the  numbers  of  these  mass-units  at  the  second 
period,  or  if  we  use  the  geometric  mean  with  weighting  aeconliug  to  the  geometric 
means  of  the  numbers  of  these  mass-units  at  both  periods  (whenever  it  happens 
that  these  are  equal),  all  these  means  (and  the  first  two  averages  in  all  cases) 
always  give,  applied  to  the  same  cases,  identically  the  same  results,  (and  in  the 
more  complex  cases  the  g<;ometric  average,  with   its  weighting,  generally  gives 


SCHEMATA  289 

The  only  system  of  Mciulitint;-  recjuired  by  this  ar<j^unieiit  fr(»nj 
compensation  by  equal  ([iiantities  is  the  system  of  weighting  ac- 
cording to  the  constant  sums  devoted  to  purchasing  each  class 
at  both  ])eriods.  There  is  no  use  claiming  that  because  the 
niass-(juantities  would  then  be  different,  we  ought  to  correct  this 
difference  in  the  two  worlds  com})ared  by  reducing  the  mass- 
([uantities  either  by  taking  only  the  smaller  (piantity  in  any  class 
at  either  period  (so  as  to  get  the  largest  ipiantity  common  to 
both  the  periods)  or  by  taking  some  average  of  them.  h'or 
then  the  sums  paid  for  such  reduced  or  averaged  constant  ([uan- 
tities  would  be  different,  and  the  two  worlds  would  be  no  more 
alike  than  before.  The  truth  is,  we  are  now  engaged  in  meas- 
uring the  variation  in  the  purchasing  power  (exchange-value) 
of  given  sums  of  money  by  the  variations — not  of  prices  (ex- 
cept as  these  indicate  the  others) — but  of  the  mass-quantities 
purchased.  Hence  the  w^eighting  is  not  to  be  according  to 
the  mass-quantities,  but  according  to  the  sums  devoted  to  pur- 
chasing them.  The  economic  worlds  we  arc  considering  are 
really  made  up  of  these  sums,  which  are  sujiposed  to  be  the 
same  at  both  periods ;  and  what  we  are  measuring  is  the  varia- 
tion in  their  purchasing  powers.  In  the  next  Chapter,  when 
we  have  under  examination  the  argument  from  compensation  by 
equal  sums,  the  economic  worlds  will  be  made  up  of  the  mass- 
quantities  (but  still  to  be  conceived  as  exchange-values),  the 
variables  then  being  the  sums  they  will  command,  indicative  of 
their  varying  exchange-values  in  money  and  of  money's  inversely 
varying  exchange-values  in  them. 

Here,  then,  the  weighting  being  according  to  the  constant  sums 
expended  on  each  class  at  both  periods,  it  is  only  the  first  of  the 
above  schemata  in  which  even  weighting  can  be  used.      In  the 

/      2  —  a,'\ 
second  the  weighting  is  1  for  [A]   and  ^  I  =  >  "'   )  ^**i"  [J>]j 

very  nearly  the  same  results  as  the  other  two  with  theirs).  Here  in  the  first  two 
averagings  the  periods  of  the  weightings  are  inverted  from  those  given  as  general 
principles  in  Chapt.  VIII.  Sect.  I.  §  7.  This  is  because  tlie  variations  of  the 
mass-quantities  are  the  inverse  of  the  price  variations.  A  similar  identity  will 
occupy  our  attention  in  the  next  Chapter  (where  the  periods  of  the  weiglitings 
are  the  proper  ones,  and  the  weighting  itsalf  is  proper).  At  present  we  have  little 
interest  in  these  relations. 

19 


■JIM)  TIIK    MI-:ril(»l)    F()l{    CONsTAN'r    SIMS 

and  ill  the  rliifd  it  is  1   for  [A]   mid  ^,1=      ,  I  f<»i"  [!>]•      Now 

if  we  use  the  liarinonic  avora^e  on  each  of  these  cases  with 
these  weightings,  we  again  always  get  an  indication  of  constancy. 
But  if  we  use  the  other  averages  with  these  weightings,  we  get 
very  ditfc  rent  results.  Tiie  arithmetic  average,  in  each  case 
with  its  ])ro})er  weighting,  indicates  for  the  first  a  rise  of  12^ 
j)er  cent.,  for  the  second  a  rise  of  25  per  cent.,  and  for  the  third 
a  rise  of  K)^  per  cent.  And  the  geometric  average,  in  each 
case  with  its  proper  weighting,  indicates  for  the  first  a  rise  of 
6.066  })er  cent.,  for  the  second  a  rise  of  14  ])cr  cent.,  and  for 
the  third  a  rise  of"  H.44  per  cent. 

As  neither  of  these  other  two  systems  pretends  to  (4aim  con- 
stancy when  the  ])roper  weighting  is  used  with  each  average,  it 
is  not  easy  to  compare  the  different  averages  on  these  schemata. 
To  c()ni})are  them  we  shall  want  rather  the  schemata  in  which 
equal  constant  sums  are  spent  ou  every  class,  so  that  even 
weighting  may  be  used  in  each  case,  wherefore  each  average,  the 
|)rice  variations  being  as  before,  will  indicate  constancy. 

>?  .').  Let  us  now  notice  the  same  }»rice  variations  with  (>(pial 
numbers  of  mass-uuits  gained  and  lost  when  these  are  eqaivd- 
lent  at  flic  second  period.  It  is  easy  to  adapt  the  preceding 
.schemata  to  these  new  mass-units.  We  need  to  change  only  the 
mass-unit  of  [1>],  which  we  siiall  represent  by  1>'.  It  nmst  be 
remarked  that  we  are  not  rearranging  the  presentation  of  the 
same  facts  as  in  the  previous  schemata,  but  we  are  presenting 
different  facts.     The  schema  for  the  hnruKtuic  ])riee  variations  is 

I   KKI     \  (>i    1.00      100     B'  C'(    2.00— 1*00  I  100  for  [A]      200  for  [15], 
II     (jC.n  A  Oi   1.50     ]83,A  IV  0>    l.oO  — 2(»0  |  100  for  [A]     20U  for  [1!]; 

that  for  the  arithmetic  ])riee  \ariations, 

I  100    A  r-r  1.00     ;«^  IV  Oi   .S.OO  — l:«.i  I  lOO  for  [A]      lOO  for  [B], 
11     W>i  A  (»    l.hO     Mz  W  ("    l.rwi  — l:];!;,   I   100  for  [Al      lOO  for  [!'.]; 

that  foi'  the  geometric  price  variations, 

1  100    A  Or.  1.00       (lOJi  B'  ('I  2.2.J—  lGC>s  I  100  for  [A]     150  for  [B], 
1!     Mi  \  0,   1..10     100    WO,    ]..-)()  — KlOJi  I  100  for  [A]     150  for  [B]. 


SCJIKMATA  291 

Here  the  varviiii>-  numbers  of  the  mas.s-unit.s  are  the  same  as 
in  the  previous  schemata,''  hut  their  prices  beino-  variousl}' 
higher  in  each  case,  the  constant  sums  expended  on  [B]  are  in 
each  case  variously  different.  That  the  above  noticed  peculiar- 
ities in  regard  to  the  numbers  of  the  mass-units  in  this  rear- 
rangement are  univ'ersal,  could  easily  be  j)roved  in  a  manner 
similar  to  that  before  used,  by  finding  first  the  gcnei-al  formula 
for  the  cases  ^vhen,  ^9,  equalling  «.„  ^i,  is  the  figure  that  has 
varied  harmonically,  arithmetically  or  geometrically,"  and  apply- 
ing this  as  before. 

Here  again  the  harmonic  average  applied  to  the  first  case,  the 
arithmetic  applied  to  the  second,  the  geometric  aj)plied  to  the 
third,  each  'with  even  weighting,  all  indicate  constancy  of  the 
general  exchange-value  of  money.  But  in  this  schematization 
it  is  only  the  second  case,  in  which  the  price  variations  are 
arithmetic,  that  has  a  right  to  the  use  of  even  weighting.  In 
the  first  case  the  weighting  is  1  for  [A]  and  2  for  [B] ;  and 
with  this  weighting  the  harmonic  average  indicates  a  fall  of 
prices  by  10  per  cent.  And  in  the  third  the  weighting  is  1  for 
[A]  and  H  for  [B];  and  with  this  weighting  the  geometric 
average  indicates  a  fall  of  prices  by  7.79  per  cent.  But,  always 
used  with  the  proper  weighting,  the  arithmetic  average  indicates 
constancy  in  every  case.  Therefore  also  these  schemata  are  not 
suitable  for  comparing  the  averages  ;  and  when  we  readapt  them 
all  to  the  same  (even)  weighting  only  the  second  will  remain. 

§  4.  Lastly  we  ^v'mh  to  schematize  the  compensation  by  equal 
mass-quantities  when,  in  the  same  price  variations,  the  mass- 
units  are  equivalent  over  both  the  periods  foc/efJier.  As  A  is  sup- 
posed in  all  the  cases  to  rise  from  1.00  to  l.oO,  its  (geometric) 
mean  price  is  always  1.2247.  Therefore  the  (geometric)  mean 
price  of  B"  is  desired  always  to  be  1.2247.  This  is  ol)tained 
in  the  following  schemata — for  the  harmonic  price  variations  : 

5  Hence  what  is  stated  in  the  last  note  is  still  iipplieable  also  to  the  i-ases  with 
these  mass-units.  And  the  numbers  of  these  mass-units  are  no  better  criteria  of 
weighting  than  the  numbers  of  the  mass-units  there  used. 

•^  This  formula,  with  «/  at  1.00  and  with  a  taken  as  the  unit  sum,  is 


in  which  f3.,  =  aJ . 


292  TIN-:    MKI'IKH)    Foil    (ONSTA  NT    SIMS 

r  100    A  @  1.00     100    B''  @  1.4142  — -JOU  I  100  tor  [A]      141.42  f.ir  [R], 
II    OG'i  A  ('(}  l.-JO     133^  B'^  @  l.OOOO  —  200  |  100  for  [A]     141.42  for  [R], 

(in  which  1.4142  :  1.0G06  ::  1.00  :  0.75,  and  1.4142  x  l.()0O() 
=  1.00  X  l.oO)  ;  for  the  aritlimctic  ])ricc  variations: 

I  100    A  @  1.00     33^  W  (a)  1.7:^2  —  133,1  I  lOO  for  [A]     .J7.73  for  [B], 
II    66|  A  @  l.oO     661  B''  (d,    .866  — 133J  |  100  for  [A]     57.73  for  [B], 

(in  which   1.732  :  O.SIK;  ::  1.00  :  O..")0,  and   1.732  x  O.S()6  = 
1.00  x  l.oO)  ;  for  the  geometric  })riee  variations  : 


I  100    A  (a)  1.00       66j  B''  %  1.50  —  166s 
H    66|  A  @  1.50     100    B"  R.  1.00-166;:; 


100  for  [A]     100  for  [B], 
100  for  [A]     100  for  [B]. 


Here  also  the  numbers  of  tlie  mass-units  are  the  same  in  the.se 
still  other  circumstances  as  in  the  })recedino;  two  sets  of  sche- 
mata ;  "^  but  with  still  other  constant  sums  expended  on  [A]  and 
[B] .  By  the  same  method  of  proof  the  above-noticed  peculiar 
relations  between  these  numbers  can  be  proved  to  be  universal.* 
Of  course,  as  before,  with  even  weighting,  the  harmonic  average 
of  the  j)rice  variations  in  the  first  case,  the  arithmetic  average 
of  them  in  the  second,  and  the  geometric  mean  of  them  in  the 
third,  all  indicate  constancy.  But  here  it  is  only  in  the  third 
that  even  weighting  is  proper.  In  the  others,  each  average 
with  its  proper  weighting,  gives  a  various  result — the  harmonic 
in  the  first  indicating  a  fall  of  prices  by  5.41  per  cent.,  and  in 
the  second  the  arithmetic  indicating  arise  of  prices  by  l.'>.-">3 
per  cent.  Therefore  again  w(!  must  re-adapt  the  schemata,  only 
the  third  here  given  being  serviceable.  It  may  be  added  that 
in  the  other  two  cases,  with  their  proper  weighting,  the  geo- 
metric average  yields  results  indicating  slight  divergences  from 
constancy  (in  the  first  a  fall  of  prices  by  0.06  ])er  (icnt.,  in  the 
second  a  rise  by  0.3  per  cent.).  The  reason  for  these  diver- 
gences is  already  known.      Tlicir  lucaning  will  Ix' explained  later. 

■^  Hence  again  the  statement  in  Note  4  is  applicable  also  to  tiie  cases  witli  f/itsr 
mass-units.  But  now  the  numbers  of  these  mass-units  are  good  criteria  of  wciglit- 
ing,  as  we  shall  see  i)resently. 

*  The  general  formula  for  the  relation  between   the  sums,  still  with  «,'  at  Hid 

ami  with  a  as  the  unit  sum,  is 

a.,'        1 

b'  =    o JT' 

I'l       Pi 

in  whieii  (iiij  —  n.,' . 


TiiK  (;eu.mktkic  mean    provkd 


293 


II. 

§  1.  When  the  argunu'iit  from  compensation  l)y  equal  mass- 
quaiitities  i.s  made  an  argument  especially  for  the  harmouic 
average  of  price  variations,  and  the  harmonic  price  variations, 
that  are  supposed  to  represent  constancy  are  contrasted  with  the 
other  variations  in  a  manner  detrimental  U)  the  argument  for 
them,  we  have,  for  the  simplest  cases,  which  use  even  weighting 
throughout,  the  following  schemata — for  the  harmonic  price 
variations  : 


r  100    A  (aj  1.00     100    B  (Tf,  1.00—200 
II    66tAr«j,l.oO     rSSiB(7r,    .75—200 

for  the  arithmetic  price  variations  : 

I  100    A  (^  1.00     100    B  (o.  1.00—200 
II    66 «  A  fj,.  1.50     200    B  (^j,    .50—2665 

for  the  geometric  price  variations  : 

I  100    A  ^0  1-00     100    B  (tr  1.00  —200 
II    661  A  @/ 1.50     150    B@    .66t— 216f 


100  for  [A] 
100  for  [A] 


100  for  [B], 
100  for  [B]; 


100  for  [A]     100  for  [B], 
100  for  [A]     100  for  [B]; 


100  for  [A]     100  for  [B], 
100  for  [A]     100  for  [B]. 


Here,  the  mass-units  all  being  equivalent  at  the  first  period, 
there  is  compensation  by  ecjual  mass-quantities  (equal  numbers 
of  these  mass-units)  only  in  the  case  of  the  harmonic  price  vari- 
ations, so  that  it  is  only  in  this  case  that  the  purchasing  power 
or  exchange- value  of  money  seems  to  be  constant,  wherefore  the 
harmonic  average  seems  to  be  the  right  one,  as  it  alone  indicates 
constancy  in  this  case;  while  in  the  other  price  variations  the 
compensation  by  mass-quantities  is,  in  the  arithmetic  price  vari- 
ations, for  every  one  third  lost  on  [A]  a  gain  of  one  whole  on 
[B] ,  which  seems  very  much  too  much,  and  in  the  geometric 
price  variations  for  every  loss  of  one  third  on  [A]  a  gain  of 
one  half  on  [B],  which  still  seems  too  much,  wherefore  the 
arithmetic  and  the  geometric  averages  or  means  seem  to  be 
wrong  because  each  in  its  own  case,  with  the  proper  weighting, 
indicates  constancy.  It  is  wholly  and  solely  on  account  of  this 
special  arrangement  of  the  mass-quantities,  due  to  the  selection 
of  the  mass-units,  which  are  equivalent  at  the  first  period,  that 


•2;»4  11  ir.  MirnioD   for  constant  sims 

some  persons  have  Ix'cn  led  to  suggest  the  lianuoiiic  average  of 
price  variations  as  the  right  one. 

But  it  is  precisely  because  of  this  special  arrangement  tliat 
tiie  harmonic  average  may  be  proved  to  be  wrong. 

§  2.  The  argument  for  the  harmonic  average  claims  that, 
the  classes  [A]  and  [B]  being  constantly  e((ually  large  oi-  im- 
[>ortant  over  both  the  i)eriods,  if  A  and  B,  the  equivalent  mass- 
units  at  the  first  period,  are  equally  precious  at  the  first  period 
(being  equally  heavy  or  bulky),  a  loss  of  one  third  on  [A]  is 
correctly  compensated  by  a  gain  of  the  same  mass  (by  weight  or 
bv  bidk)  on  [B] ;  or  if  they  are  not  equally  ])recious  at  the  first 
[)criod,  a  loss  of  one  third  on  [A]  is  correctly  compensated  by 
a  gain  in  the  same  proportion,  namely  by  one  third,  on  [B], 
this  addition  of  ^  B  being  as  much  lai-ger  or  smaller  than  ^  A 
as  B  was  less  or  more  precious  than  A  at  (lie  first  period.  Evi- 
dentlv  there  is  here  a  tacit  assumption,  which  is  belied  by  the 
verv  supposition  itself,  of  continuance  at  the  second  ])eriod  of 
the  same  relative  preciousness  as  at  the  first  period.  A  and  B 
are  sup])osed  to  be  equivalent  at  the  first  ])eriod.  Then  ^  A  and 
J  B  are  equivalent  at  the  first  period.  Therefore  at  the  first 
period  if  we  distribute  our  purchases  so  as  to  get  with  two 
monev-units  §  A  and  1 J  B  instead  of  1  A  and  1  B,  we  have  per- 
fect compensation  in  the  gain  of  J  B  making  up  for  the  loss  of 
^  A,  because  this  gain  is  equivalent  to  this  loss.  But  at  the 
second  period  the  supposition  is  that  A,  having  risen  in  price  to 
1.50,  while  B  has  smik  to  .75,  has  become  more  valuable  than 
B — in  fact,  just  twice  as  valuable  in  this  example.  Therefore 
J  B  is  no  longer  e(piivalent  to  .'j  A,  and  the  gain  of  I  B  is  no 
longer  sufhcient  to  compensate  for  the  loss  of  I  A. 

Or  let  us  i)ut  the  (piestion  as  one  concerning  purciiasing 
power.  If  at  t\w  first  perio<l,  possessing  two  money-units,  we 
give  up  one  third  of  our  |)urchasing  ])ower  over  [A]  by  using 
only  3  M  to  ])urchase  [A],  and  if  at  the  same  time,  with- 
out any  intervening  changes  of  prices  we  gain  one  third  in  our 
purchasiixg-  power  over  [B]  by  using  the  ^-  M  saved  from  use 
on  [A],  tlicHMs  perfect  conq)ensation  ;  for  the  counter-bahuicing 
purchasing  powers  are  equal.     But  the  siq)position  we  are  deal- 


THE    GKOMKTItlC    MKAN    I'UOVKD  295 

ing  with  is  tluit  at  the  second  period  we  can  jiet  oul\'  H  .V  with 
1  M,  and  it  is  chiinied  that  the  compensation  is  good  if  wo  can 
get  1^  B  witli  the  other  1  M.  Xow  nnder  these  eireunistances 
it  is  true  that  the  particular  exchange-vahie  of  M  in  [A],  or  its 
particuhir  purchasing  power  over  [A],  lias  fallen  bv  one  third, 
and  that  the  particular  exchange-value  of  M  in  [B] ,  or  its  par- 
ticular purchasing  power  over  [B] ,  lias  risen  by  one  third,  and  so, 
if  we  were  dealing  with  ]>articular  purchasing  powers,  this  com- 
pensation might  appear  to  be  good.  But  we  are  really  dealing 
with  the  general  exchange- value  of  M,  and  so  \\ith  its  general 
purchasing  power.  This  has  not  fallen  by  one  third  because  of 
M's  purchasing  only  f  A,  unless  M  also  purchases  only  |  of 
everything  else  ;  nor  has  it  risen  by  one  third  because  of  M's 
purchasing  1^  B,  unless  M  also  purchases  1^  of  everything  else 
— \vhich  conditions  are  contradictory  to  each  other  and  to  the 
original  supposition.  AVe  do  not  then  as  yet  know  what  the 
compensation  ought  to  be.  But  ^ve  do  perceive  this,  that  when 
A  alone  rises  in  price  and  B  alone  falls  in  ])rice,  M  in  gaining 
^  in  ([uantity  of  [B],  which  has  fallen  in  exchange-value,  has 
gained  less  in  exchange-value  than  it  has  lost  in  losing  ^  in 
quantity  of  [A],  which  has  risen  in  exchange-value.  The  com- 
pensation offered  is  no  longer  an  addition  of  one  third  of  an 
equal  exchange-value  or  purchasing  power  in  ])lace  of  a  sub- 
traction of  one  third  of  a  given  purchasing  jiower  ;  it  is  the 
addition  of  one  third  of  a  smaller-grown  purchasing  power  in 
place  of  a  subtraction  of  one  third  of  a  larger-grown  purchas- 
ing power. 

In  such  cases,  therefore,  our  money  has  fallen  in  j)urchasing 
power  or  exchange-value — it  has  dejn-eciated.  And  inversely 
the  general  level  of  prices  has  risen.  The  fall  in  the  price  of 
B  to  the  harmonic  term  (a  fall  by  |)  is  not  great  enough  to 
compensate  for  the  rise  in  the  price  of  A  (a  rise  by  |).  And 
the  harmonic  average  of  the  price  variations  is  wrong  in  indicat- 
ing constancy. 

The  fallacy  in  the  argument  for  the  liarmonic  average  of  prices 
from  com])ensation  by  equal  mass-quantities  is  that  it  employs 
a  comjiensation  which  is  good  only  on  the  assumption  that  the 


2^)(]  Till".    MHTIIOl)    I'Oll    CONSTANT    SUMS 

r('l;>tinnslii|)  lictwccii  the  cxcIkuiu'c-n'mIiU's  or  prociuiisiu'ss  ot"  the 
nrticlo  li;is  rcmiiiiicd  the  muiic  :it  the  sccuiul  as  at  the  tii'-t  period, 
altlioiiti-h  the  data  ar<i;ued  upon  preehide  tliis  eoiitiiiiiimee  (ex- 
<'ept  ill  ease  there  are  no  irreytilai'  |)riee  variations). 

§  .").  ( )ii  the  other  hand  when  the  ariiiunent  I'roni  eoinpensa- 
tion  bv  equal  mass-(|uantities  is  ap|)lied  to  the  arithmetie  aver- 
age of  priee  variations,  there  is  a  siinihir  falUicy  reversed.  The 
aro-unient  now  tacitlv  assumes  that  the  conditions  existinji:  at  the 
second  |)eriod  existed  also  at  the  first  ;  for  it  eiupl(»ys  the  coin- 
])(!nsation  which  is  g'ood  only  at  the  second  ])eriod. 

Thus  in  the  above  schema  for  the  arithmetic  price  \ai'iatioiis, 
where  A  and  I>  are  (M|uivalent  at  the  first  period,  the  compensa- 
tion offered  is  a  ^ain  of  one  whole  A  for  every  one  third  B  lost, 
the  gain  being'  three  times  as  large  as  the  loss,  measured  in  those 
mass-units.  But  we  perceive  that  at  the  second  ])eriod  when 
the  j)riee  of  A  lias  risen  to  l.oOand  the  price  of  B  fallen  to  .50, 
A  has  come  to  be,  at  the  second  i)eriod,  three  times  as  valuable 
as  B,  wherefore,  at  the  second  period  itself,  a  compensation  by 
a  gain  of  three  times  as  much  of  B  as  is  lost  of  A  is  the  proper 
compensation  then.  At  this  second  |)erio(l  we  are  compensated 
by  getting  just  as  many  times  as  much  more  of  the  fiillcii  article 
than  less  of  the  risen  as  the  former  has  become  less  valuable  than 
the  risen.'  Here  is  a  semblance  of  correctness  in  the  jmsition  of 
the  arithmetic  averagist.  The  semblance  is  brought  out  more 
plainly  by  the  following  rearrangement  of  the  schemata,  in  which 
merely  the  mass-units  of  [B]  are  altered,  and  consccpiently  the 
numbers  of  them  pui-cliasa])le  with  the  same  sums  of  money,  the 
fjicts  represented  being  the  same  as  before.  The  schema  fi»r  the 
harmonic  pri(;e  variations  is  : 

1100    A  r^  1.00     .")()     ly  (/^  2.00  — !.')()     I  100  for  [A]      100  for  [I!], 
II    66|  A  @  I.'jO     00^  B'  (a)  \.r^()—i:V.]ls  I   100  for  [A]      100  for  [15]; 

that  for  the  arithmetic  |)rice  variations  : 

1  100    A  (W)  1.00     88|  IV  (h)  8.U0  —  V.V^ 
II    66|  A  @  l.oO     66|  B^  0,)  1.50  —  183} 


100  fur  [A]      100  for  [H], 
100  for  [A]     100  for  [B]; 


*  Tlie  universality  of  this  relationship,  given  eonditions  permitting  of  even 
weighting,  is  evident  when  we  renieml)er  that  the  mass-quantities  are  in  harmonie 
progression  ;  for  in  this  progression  around  unity  as  the  mean  we  know  that 
\     a:h      \  :  :  a:  b. 


rilK    (JKoME'riMC    MKAN     I'ltONKD  297 

tliat  for  tilt'  tro(MiH'tric  price  variations  : 

I  100    A  Or,  1.00     44|  IV  0<j  2.25  —  1-14|  I  100  for  [A]     100  fur  [B], 
11    6(i|  A  (^,  l.oO     (i6|  B'  Oij  1.50  —  133i  |  100  for  [A]     100  for  [B] ; 

among  which  it  is  only  the  arithmetic  price  variations  that  gives 
compensation  hy  arithmetically  eqnal  nnmbers  of  these  mass- 
units  ;  while  in  the  harmonic  price  variations  the  compensation 
seems  to  be  very  much  too  small  (although  it  is  really  the  same 
as  before,  and  proper  at  the  first  period  itself),  and  in  the  geo- 
metric still  too  small ;  wherefore  now,  by  a  mere  change  in  the 
size  of  the  mass-units,  the  arithmetic  average  of  the  price  varia- 
tions, indicating  constancy  in  this  case,  alone  seems  to  be  justi- 
fied in  indicating  constancy,  and  so  seems  to  be  the  pro j)er  aver- 
age to  use  in  all  cases. 

The  fault  with  this  argument  for  the  arithmetic  average  of 
price  variations  is  that  the  compensation  by  mass-quantities 
which  it  offers  is  what  ought  to  take  place  at  the  xccond  period 
alone.  At  this  period  when  A  =  B'  (or  when  A  =  3  B),  in  an 
even  spending  of  three  money-units  on  [A]  and  [15]  we  get 
1  A  and  1  B'  (or  -">  B),  and  again  in  a  s})ending  of  one  third  less 
on  [A]  and  of  one  third  more  on  [B]  we  gain  just  as  much  as 
we  lose  ;  for  we  gain  ^  B'  (or  1  B)  in  place  of  ^  A  lost.  Here 
the  compensation  is  perfect  because  the  quantity  lost  is  equiva- 
lent to  the  quantity  gained  in  both  the  transactions  compared 
(whether  the  masses  gained  and  lost  happen  to  be  expressed  in 
e(|ual  or  unequal  numbers  of  mass-units,  according  to  the  sizes 
of  these).  But  in  our  suppositional  case  we  are  comparing  a 
transaction  at  the  first  period  when  A  (being  equivalent  to  B) 
was  equivalent  only  to  J  B',  with  a  transaction  at  the  second 
period,  at  which  alone  A  is  equivalent  to  B'  (or  to  3  B).  An 
offered  compensation  of  ^  W  (or  1  B)  for  a  loss  of  J  A  at  the  first 
period  would  be  three  times  too  great.  It  is  still  too  great, 
though  not  so  much  in  excess,  when  it  is  offered  at  the  second 
period  in  comparison  with  the  first.  We  must  reflect  that  ^  B' 
(or  1  B)  has  fallen  in  price,  that  is,  it  was  more  valuable,  and 
that  A  has  risen,  that  is,  it  was  less  valuable.  The  gain  of  a 
numerically  equal  amount  (or  three  times  as  much)  on  an  article 
which,  although  its  iliass-miit  is  equally  (or  three  times  less)  valu- 


21>S  Tin-:  MKTiioi)  von  cosstast  sfms 

able  than  the  other  at  the  sceond  pcriocl,  was  three  times  ni(»re 
(or  equally)  vahiahle  at  the  first  period,  is  too  ,i2:rcat  a  li-aiii. 
Similar  would  be  the  conelusion  if  we  treated  the  subject  from 
the  point  of  view  of  purchasino-  power.  Therefore,  as  this  prof- 
fered compensation  is  too  large  for  the  (piantity  gained  over  the 
(juautitv  lost,  our  money  purchases  too  much  to  ])erniit  it  to  be 
stable  :  its  [)urchasino-  power,  and  its  exchange-value,  ha^  rist-n  : 
it  has  a])j)reciated.  And  inversely  the  arithmetic  compensation 
bv  equal  prices  is  too  great  for  the  loss  over  the  gain,  and  the 
general  level  of  prices  has  fallen,  instead  of  being  constant,  as 
is  wrongly  indicated  by  this  average. 

The  generalization  may  therefore  be  made  that  in  all  cases 
when  constant  sums  are  spent  on  the  classes  at  l)oth  periods, 
these  sums  being  taken  for  the  weights,  the  indication  concern- 
ing the  general  level  of  prices  olfered  by  the  harmonic  average 
of  the  price  variations  is  lower  than  it  ought  to  be ;  and  the  in- 
dication offered  by  the  arithmetic  average  of  the  price  variations 
is  higher  than  it  ought  to  l)e. 

§  4.  The  true  position  must  take  account  of  the  conditions  at 
both  the  periods,  and  so  the  compensation  must  lie  between  those 
offered  by  the  harmonic  and  the  arithmetic  terms,  being  larger 
than  the  former  and  smaller  than  the  latter  for  the  h)ss  over  the 
gain  by  mass-cpiantities.  It  is  herc^  that  lies  the  compensation 
offered  by  the  geometric  mean  and  average. 

The  schemata  to  illustrate  the  argument  for  the  geometric 
mean  are  as  follows — for  the  harmonic  price  variations  : 

I  100    A  0>    1.00     70.71  B''  Oi\  1.4142  —  170.71  1  100  for  [A]      KH)  for  [P.], 
II    ()«!  A  Or,  l.oO     04.28  V/'  Or,  1. 0(500  —  100.04  |  100  for  [A]      100  for  [P,]; 

for  the  arithmetic  price  variations  : 

I  100    A  («,  1.00       .")7.7.']  W  (.<■  1.732  —  lo7.73 
II    00|  A  («   l..->0     11.-). 47  W  ('<>    .800  —  182.18 

for  the  geometric  price  variations: 


100  for  [A]      100  for  [15], 
100  for  [A]      100  for  [P.]; 


I  100    A  ("\  1.00       0()^  W  ('i\  1..J0—  1()0^'  I  KM)  for  [A]      ino  for  [P.], 
II    6G|;  A  (<i    l.od     10(1     W  <<!•  1.00— KUlj  |  100  for  [A]      100  for  [!'.]• 

Here,  the  real  facts  represented  being  the  same  as  in  the  two 
preceding  sets  of  sclien)ata,  the  only  compensation  l)y  an  equal 


THE     (rEOMETIlIC    MEAN     1'1{()VE1)  299 

number  of  fhrsc  mass-units  is  in  the  simple  geometric  price 
variations,  the  compensation  offered  by  the  harmonic  price 
variations  being  too  small,  and  that  by  the  arithmetic  too  large. 
Thus  we  have  the  appearance,  equally  good  as  such,  that  it  is 
only  the  geometric  mean  of  price  variations  that  in  indicating 
constancy  in  this  case  gives  the  right  result.  The  (piestion  now 
arises.  Are  there  any  reasons  \vliy  the  appearance  is  better  in 
this  case  than  in  the  others  ? 

The  reasons  are  plain.  They  flow  from  the  principles  already 
examined  in  Chapter  VIII.,  which  principles  are  of  direct  ap- 
plication to  our  present  subject.  AVe  are  dealing  ^vith  a  subject 
in  which  no  matter  how  far  the  price  of  A  rises,  the  quantity  of 
[A]  purchasable  with  a  given  sum  cannot  fall  to  /.ero,  and  as  B 
falls  in  price,  the  quantity  of  [B]  can  be  limited  by  no  figure 
short  of  infinity,  (^ur  subject  then  is  suitable  for  the  use  of  the 
geometric  averaging  of  the  price  variations,  in  which  constancy 
is  shown  when  the  quantities  offered  in  compensation  vary  to 
the  geometric  extremes.  Here  the  variations  are  all  the  reverse 
of  each  other.  The  mass-unit  of  [B]  is  no  longer  the  equivalent 
of  the  mass-unit  of  [A]  either  at  the  first  period  or  at  the  second 
period,  but  it  at  the  second  period  is  equivalent  to  A  at  the  first 
and  it  at  the  first  is  equivalent  to  A  at  the  second.  And  in  tlie 
variations  of  the  numbers  of  the  mass-units  there  is  compensation 
not  only  by  equal  quantities  but  by  equality  of  distance  traversed 
over  the  same  road  in  reversed  directions.  These  inverted  re- 
lations are  universal  (for  all  simple  geometric  price  variations  of 
two  classes  in  opposite  directions)  when  we  have  the  conditions 
required  by  even  weighting.  It  f  )llows  from  them  that  there  is 
alternation  of  preciousness  conjointly  with,  and  op])ositely  to, 
the  alternation  of  the  mass-quantities.  We  lose  ojie  third  on 
the  whole  of  [A]  while  it  gains  in  preciousness  by  half  (reckon- 
ing from  the  first  period),  and  (reckoning  in  the  same  Avay)  we 
gain  one  half  on  the  whole  of  [B]  while  it  loses  in  preciousness 
by  one  third.  This  leads  to  the  really  fundamental  reason, 
which  is  : — As  the  mass-unit  here  used  of  [B]  is  the  equivalent 
of  the  mass-unit  of  [A]  over  both  the  pcrioih,  it  is  obviously  cor- 
rect that  we  should  gain  as  many  of  these  mass-units  of  [B]  as 


300  IIIK    MKTllOD    FOU    CONSTANT    SIMS 

we  lose  of  these  iiiass-uuits  of  [A]  .      The  eoiiipensatioii  by  iiiass- 
qiiantities  is  hi  this  ease  perfect. 

The  superiority  of  the  o-eomctrie  mean  to  the  other  moans  of 
])riee  variations  may  be  shown  more  clearly  l)y  bringing  together 
the  three  forms  of  the  schemata  on  which  the  argument  for  each 
of  the  averages  has  r(>lie(l.  These  are — for  the  harmonic  average 
of  price  variations  : 


I  lOU    A  ('S'  l.UO     100    B  ('J   1.0(»  — 200 
II    06|Af'rl.50     \53\Bfa,    .75  —  200 


100  for  [A]      100  for  [B], 
100  for  [A]     100  for  [B]; 


for  the  arithmetic  average  of  price  variations: 

I  100    A  Oi   1.00     33i  B^  @  3.00  — 133i  1  100  for  [A]     100  for  [B], 
II    ()62  A  Or.  1.50     6(5|  B'  ("■  1.50  — 133^  |  100  for  [A]     100  for  [B]; 

for  the  geometric  mean  of  price  variations  : 


T  100    A  ('>  1.00      66|  B''  @  1.50  — 166| 
II    661  A  Of.  1.50     100    B''  (7f)  1.00  — 166| 


100  for  [A]     100  for  [B], 
100  for  [A]     100  for  [B]. 


In  the  first  of  these  it  is  plain  that  the  mass-miit  used  of  [B] 
is  of  smaller  exeluinge- value  than  the  mass  unit  of  [A]  over 
both  the  [)eriods  together.  Hence  in  gaining  only  an  equal 
number  of  these  mass-units  of  [B]  fen*  the  mass-units  of  [A] 
lost,  we  gain  less  exchange-value  than  Ave  lose.  Therefore  our 
money  has  diminished  in  exchange-value  or  purchasing  power, 
or  has  deprecnated,  and  the  general  level  of  prices  has  risen  ; 
wherefore  the  harmonic  average  of  the  price  variations  errs  be- 
low the  truth  in  indicating  constancy. 

In  the  second  it  is  plain  that  the  mass-unit  used  of  []>]  is  of 
greater  exchange-value  than  the  mass-unit  of  [A]  over  both 
the  periods  together.  Hence  in  gaining  an  equal  number  of 
these  mass-units  of  [15]  for  the  mass-units  of  [A]  lost,  we  gain 
more  exchange-value  than  we  lose.  Therefore  our  money  has 
augmented  in  exchange-value  or  |)urchasing  ])ower,  or  has  a[)- 
j)reciated,  and  the  general  level  oi"  prices  has  fallen  ;  wherefore 
the  arithmetic  average  of  the  j)rice  variations  errs  above  the 
truth  in  indicating  constancy. 

I)Ul  ill  the  thii'd  the  mass-unit  w^vd  of  [!>]  is  of  the  same  ex- 
change-value as  the  mass-unit  of  [A]  over  both  the  periods  to- 
gether.     Hence  in  gaining  an  (npial  number  of  these  mass-units 


THE  GEOMETRIC  MEAN  PROVED  .'JOl 

of  [B]  for  the  iiiasi^-uuiis  ol'  [.V]  lost,  we  gain  exaetly  the  same 
exchange-value  as  we  lose."  Therefore  our  money  has  remained 
constant  in  exchange- value  or  purchasing  power,  and  the  gen- 
eral level  of  prices  is  also  constant ;  wherefore  the  geometric 
mean  is  right  in  indicating  constancy. 

The  geometric  mean  also  rightly  indicates  the  variations  in 
the  other  cases.  Let  us  go  back  to  the  first  set  of  schemata  in 
this  paragraph,  which,  in  the  same  order,  represent  the  same 
states  of  things  as  the  last.  For  there  we  have  reduced  the 
mass-unit  of  [B]  to  equivalence  over  both  the  periods  with  the 
mass-unit  of  [A] .  In  the  first  schema,  where  the  price  varia- 
tions are  the  harmonic,  the  purchasing  power  of  money,  or  its 
exchange- value,  relatively  to  these  two  classes,  is  evidently  ac- 
cording to  the  totals  of  these  equivalent  mass-units  which  the 
given  total  sum  of  money  will  ])urchase  at  each  period,  so  that 

Mo,       160.94  . 

we  have  ^tf—  =     _ „  ^s  =  ().H42<S,  mdicatmg  a  tall  ot  o./2  per 

cent.,  while  the    general    price  variation    is    the   inverse,   thus 

Po       170.71  .-,...,. 

p  =  Yft7%  n  1  =  l.ObOO,  mdicatmg  a  rise  of  0.06  ])ercent.    Aow 

,,....     P..  (3       ?> 

the  geometric  mean  ol  these  price  variations  is   ^-  ^  ^      x 

=  — -  =  1.0606,  likewise  indicating  a  general  rise  of  prices  by 

l/8 

6.06   ])er  cent.      In  the  second,  where  the  price  variations  are 

.  ,         .  ,         Mo.       182.18  _     .    ,.      . 

the  arithmetic,  we  have  ^rr^^  ^  -  __  „-,  =  l.lo4<,    indicatuiir  a 
'  3Ioi       l-u.io  '  '^ 

^.^  .-  nP.>       157.7;)  .... 

rise  ot  10.4^  per  cent.,  and  p"  =  -.^^^y^T;.  =  <>.f^b60,  indicating  a 

-  These  mass-units  are  not  the  economie  individuals  described,  for  the  present 
supposition,  in  Chapter  IV.  Sect.  V.  ^3,  namely  constant  exchange- values  with 
variable  masses.  Rut  they  are  substitutes  therefor  ;  and  they  are  like  the  eco- 
nomic individuals  described  in  Chapt.  IV.  Sect.  V.  j^  0,  which  will  be  used  in  the 
supposition  to  be  treated  of  in  the  next  C'hapter.  They  may  very  properly  be 
used  for  the  purpose  they  are  here  put  to.  To  be  sure,  the  weighting  of  the  classes 
at  each  period  cannot  be  measured  by  the  numbers  of  these  mass-units  they  con- 
tain. Yet  the  weighting  of  the  classes  in  the  averaging  of  their  price  variations, 
over  two  periods  together,  is  according  to  the  geometric  means  of  the  numbers  of 
such  mass-units  they  contain  at  each  period.  Another  example  will  be  given 
later  (see  below  in  Sect.  IV.  Note  1). 


302  'I'lIF,    MK'llIoD    I'oi;    CONSTANT    SIMS 

fall  of  I."). 40  j)oi"  coat.;  and   the  licoiiietric   moan   of  the   prico 

variations  is  p"  =      I -^  x  ,)  =  ~:y~  =0.8()<iO, likewise  indicating 

a  general  tall  of  prices  hy  1:').40  per  cent.  That  the  geometric 
mean  of  the  price  variations  of  two  classes  equally  important 
over  both  the  j)criods  universally  agrees  with  the  inverse  of  the 

indication  for   .,  '  rendered  hy   the   nuinhers  ol  the   niass-miits 

equivak'ut  over  hoth  the  [periods  ])urchasal)le  at  each  period,  has 
already  been  demonstrated  near  the  end  of  the  preceding  Chap- 
ter in  fornuda  (10)  there  given. '^  Therefore  the  geometric  mean 
of  price  variations,  whenever  it  is  applicable  to  cases  in  which 
for  two  classes  of  goods  the  same  sums  of  money  are  spent  at 
both  periods,  universally  gives  the  right  indication. 

§  5.  The  fault  with  the  argument  from  arithmetic  com- 
pensation by  e(pud  mass-quantities,  as  it  has  generally  been 
employed  by  the  writers  who  have  suggested  or  advocated  or 
employed  the  harmonic  average  of  price  variations,  has  lain  in 
the  fact  that  utter  neglect  has  been  paid  to  the  exchange-value 
of  the  mass-(|uantities  whose  loss  and  gain  have  l)een  compared^ 
To  be  sure,  the  mass-units  have  generally  been  chosen  equiva- 
lent at  the  first  period.  But  that  has  been  due  merely  to  conve- 
nience, and  to  the  habit,  itself  due  to  convenience,  of  starting  with 
units  ))riccd  at  1.00  tw  at  KH).  The  (piery  why  the  conqiensa- 
ti<»n  should  l)e  by  e(jnal  numbers  of  such  mass-units,  has  never 
been  raised  ;  wherefore  the  wrongness  of  such  conq)cnsation  has 
escapcid  notice.  Possibly  it  has  not  been  noticed  that  the  so- 
called  "  (juantities,"  or  mass-(|uantities,  as  numerical  figures,  are 
not  absolute  figures,  according  to  the  masses  given,  but  are  de- 

''  Kiirthermore,  in  liotli  tlic  aUovc  cxaniplfs,  in  wliicli  the  mass-quantity  of  [A] 
is  KHi  and  the  price  of  A  1.00,  at  tlie  first  period,  tlie  figure  for  tlic  variation  of  the 
general  exchange-value  of  money  is  identieal  with  the  hundredth  part  of  the 
mass-quantity  of  [H]  at  the  second  ])erlod,  and  also  it  is  the  quotient  of  the  price 
of  B"  at  the  first  period  divided  hy  tiie  ])rie,e  of  A  at  the  second  ;  while  the  figuri; 
for  the  general  price  variation  is  identical  with  the  price  of  B"  at  the  second 
period,  and  also  it  is  the  (luotient  of  the  price  of  A  at  the  second  period  divided 
by  the  price  of  B"  at  the  first.  The  universality  of  these  relations,  and  of  some 
others  which  it  would  he  too  long  to  notice,  is  also  demonstrated  by  formula?  (<>), 

/  If  '  \ 

(7),  and  (!M  (  and  by  their  inversions  for  ■"..      )  given  near  the  end  of  the  preced- 

iTiL'  <  liajilir. 


thp:  oeomethic  mean   i'Roved  303 

terniiued  also  by  the  sizes  of  the  mass-units  in  wliich  the  masses 
are  measured. 

Many  advoeates  not  only  of  the  harmonic,  but  even  of  the 
arithmetic  average,  have  failed,  as  we  have  seen,  at  a  still  earlier 
stage.  They  would  measure  the  general  exchange-value  of 
money,  under  the  name  of  its  "  general  purchasing  power,"  by 
the  mere  mass-quantities  that  a  given  total  sum  of  money  can 
purchase,  or  has  the  poAver  of  purchasing,  at  each  period,  with- 
out regard  either  to  the  mass-units  used  or  to  the  actual  spend- 
iugsof  our  money  at  the  two  periods  comj)ared.  Our  own  com- 
pleter analysis  has  shown  that,  when  there  is  any  change  of 
prices,  there  is  always,  within  fixed  limits,  a  great  variety  of 
total  mass-quantities  that  can  be  purchased  with  the  same  total 
Sinn  of  money,  according  as  different  spendings  of  its  parts  (the 
same  at  both  periods)  be  hit  uj)on.  Therefore  this  argumenta- 
tion proves  only  a  great  variableness  in  "general  purchasing- 
power"  so  conceived,  whenever  there  is  any  change  of  prices. 
Thus  if  we  look  u])on  purchasing  power  as  something  to  be 
measured  only  by  the  quantities  of  things  that  can  be  purchasedj 
we  see  that  for  such  a  thing  as  the  "general  purchasing  power" 
of  mone\',  so  conceived,  to  remain  stable,  absolutely  no  change 
of  prices  must  take  place.  So  far  Roscher  and  others  are  right 
in  denying  the  possibility  of  stability  of  money  in  general  pur- 
chasing power  except — to  use  Martello's  ])hrase — by  "petrifac- 
tion "  of  the  economic  world.  But  thev  are  wrono-  who  ex- 
tend  this  denial  to  the  ireneral  exchange- value  of  monev.  For 
it  is  evident  that  if  later  in  any  distribution  of  spendings  one's 
money  gets  the  same  exchange-value  which  it  got  before,  it  gets 
the  same  exchange-value  in  any  other  distribution  of  spendings. 

It  might,  however,  be  admitted  that  we  can  have  such  a  con- 
cept as  this  of  "general  purchasing  power"  measured  only  by 
quantities  of  things  purchasable,  if  we  desire  to  distinguish 
"  purchasing  power  "  from  "  exchange- value."  For,  as  just  re- 
marked, we  know  that  it  remains  constant  so  long  as  all  prices 
remain  unchanged  ;  and  we  also  perceive  that  if  all  prices  rise 
or  fall  in  exactly  the  same  proportion,  it  rises  or  fall>  in  that 
proportion — in   the  former  case  coinciding  with   exchange- value 


.304  I'lIK    MKTlIOl)    KOi;    CONsrANT    SIMS 

ill  all  other  tliiiiu's  and  witii  ('xchauiic-xaliu'  in  all  tliinus,  in  the 
latter  case  with  the  former  of  these  alone,  in  all  other  cases 
breaking  np  and  disappearino-.  Bnt  snch  an  (nancsccnt  e()neej)t 
is  of  no  service  in  economics. 

The  oidy  proper  way  to  conceive  of  ii:eneral  pni'chasinu  |)owcr 
is  by  the  qnantity  of  excliani»:e-valne  it  can  purchase.  It  would 
be  absurd  to  measure  th(>  liftino-  power  of  a  derrick  by  auother 
attribute,  say  volume,  in  the  thiniis  which  the  derrick  can  lift, 
rather  than  by  that  attribute^  in  them  which  is  the  essential  ob- 
ject to  which  liftiuii'  ])ower  has  reference,  namely  their  weight. 
If  the  volume  of  all  things  were,  and  always  remained,  of  ecpial 
weight,  we  might  theu  indeed  measure  lifting  ])ow(n-  by  the 
volume  of  the  things  lifted.  But  if  the  same  volumes  grow 
heavier,  more  lifting  power  is  needed  to  lift  them.  And  so  if 
the  same  volumes  or  weights  grow  more  valuable,  more  purchas- 
ing powder  is  required  to  purchase  them.  Therefore  it  is  equally 
absurd  to  measure  general  purchasing  power  only  by  either  the 
volumes  or  weights  of  the  things  purchasable.  The  general  pur- 
chasing power  of  a  sum  of  money  can  be  measured  only  by  the 
exchange- value  of  the  things  the  money  can  ])ur('hase.  The 
money  has  risen  in  exchange-value,  for  instance,  eitiier  if  it 
can  ])urchase  a  greater  number  of  things  that  have  the  same  ex- 
change-value as  before,  or  if  it  can  purchase  the  same  number  of 
things  that  have  a  greater  exchange-value  than  before.  The  ex- 
changc-N'alue  of  the  things  whose  quantities  purchasable  are  be- 
ing considered  must  also  betaken  into  account.  Therefore  if  we 
are  measuring  general  purchasing  power  only  by  the  constancy  or 
variation  of  the  (piantities  purchasable,  we  must  be  careful  that 
we  be  dealing  with  individual  masses  that  have  the  same  ex- 
change-value— over  both  the  ])eriods  comj)ared. 

Being  thus  identified  with  exchange-value  in  all  other  fhiiu/s, 
the  conception  of  general  ])ur(;hasing  power  can  givt'  us  no  more 
trouble — and,  of  course,  little  aid.  A\  hat  lias  caused  trouble  is 
the  fact  that  this  idea  has  been  introdnced  into  economics  as  a 
variant  iq)on  the  idea  of  general  exchange-value  more  intelligible 
than  it  and  sei"\iceal)le  to  hel])  us  in  our  conception  of  it,  or  to 
further  us  when    it   fails.      Instead,  by  introducing  comparison 


ITS    FORMULATION  305 

between  inessential  attributes  in  tilings,  this  idea  has  only  caused 
confusion,  and  it  is  responsible  for  many  of  the  aberrations  in 
our  subject.  * 

III. 

§  1.  The  measurement  of  the  constancy  or  variation  in  the 
exchange-value  of  money  by  the  numbers  of  equivalent  mass- 
units  is  not,  of  course,  confined  to  the  case  of  two  equally  im- 
portemt  classes.  It  is  as  applicable  to  any  number  of  classes, 
with  any  amount  of  unevenness  in  their  sizes,  provided  they 
satisfy  the  condition  required  for  its  application.  Therefore 
we  reach  this  general  method  of  measurement : —  When  con- 
stant sums  of  money  are  expended  at  both  periods,  find  in  all 
the  classes  masses,  to  he  used  as  units,  that  have  the  same  ex- 
change-value {or  money-value)  over  both  the  periods  together  ;  and 
measure  the  constancy  or  variatio)i  in  the  exchange-value  of  money 
by  the  constancy  or  variation  in  the  total  number  of  such  maw- 
units  purchased  at  each  period. 

If  we  have  already  executed  the  first  part  of  this  injunction, 
the  true  formula  for  the  price  variations,  in  these  cases,  is 

P  V  "   _L    J/  "      I     ~  //      I 

^  ^  'h +  ^1    +  •^i    + / 1 X 

Pi     •V'  +  i//'  +  V'  + '  ^^ 

or  a         b         c 


P2       <       ?l         H 


P,       a         b         c  ' 

a  "         C-i  "         "  " 

"2  r2  /2 

in  which,  as  before,  the  doubly  accented   letters  represent  the 
numbers  and  the  prices  of  such  mass-units  at  each  of  the  periods.^ 

*  And  yet,  as  the  term  contains  reference  to  power,  reference  to  which  is  lack- 
ing in  the  term  "exchange- value,"  the  term  "  purcliasing  power"  ouglit  to  have 
been  of  assistance.  Tliat  it  has  not  been  is  probably  due  to  the  limitation  in  the 
meaning  of  the  term  "  to  purchase  "  noticed  in  Note  4  in  Chapt.  V.  Sect.  III.  ?  8. 

1  The  second  of  these  formuhe  is  like  a  formula  cited  in  Chapt.  V.  Sect.  VI. 
§  2  as  a  miscarried  liarmonic  average  of  price  variations.  But  there  tlie  formula 
was  applied  to  any  mass-utiits,  while  here  it  is  applied  only  to  special  mass-units. 
The  weighting  here  in  this  formula,  viewed  as  a  harmonic  averaging  of  price 

a      b 

variations,  is  according  to  — „'o~7/,  ,  which  agrees  with    what  was   there 

•  ^1     Pi 
said.     Another  interpretation  of  this  weighting  will  be    noticed  presently  (for 
a  b 

;rT,  =  Xi",  0-7;  =  2/1",  and  so  on). 
"1  Pi 

20 


,30()  THK    MKIIlol)    I'oi;    (ONSI'AN'r    SIMS 

The  in;i>s-units  inav  he  of  any  sizes,  provided  they  he  e([uiva- 

leiit  over  hotli    the  pei'iods.      'Hius   in    the  examples  alxtve  used 

in  which  the  price  of  [A]  rose  hy  •")<»  i)er  cent.,  we  supposed  A 

to  be  such  that   its   price  rose  from    !.<)(»  to  l.oO,  so  that   the 

geometric  mean  price  of  this  and  of  every  e(|uivah'iit  mass-uuit 

was  1.2247.      We  mitihl  (■([iially  well  ha\'e  supposed  it>  |)rice  to 

rise  fnmi  .(JO  to  .!I0,  so  tiiat  its  geometric  mean   price  and  that 

of  every  mass-unit  eijuivalcnt    to   it  would   he  .7.S4S  ;    or  we 

might  have  supposed  any  other  rise  hy  oO  per  cent,  we  please<L 

For  such  changes  aifect  all  tlie  mass-units  and  all  the  prices  in 

the  same  j)rojiortion,  so  that  their  numbers  are  all  altered  in  the 

same  proportion  (the  inverse  of  the  j)receding),  and  conse({uently 

the  sums  of  their  miiiibers;   wherefore  the  results  are  the  same. 

a  a  r  "        d." 

%  2.   As  .(•/'  =       ,,  and  .r,/'  =      ,,  therefore  '  ',,  =    '-, ,  and 


-  ^,  =    ', ,  ;  and  similarlv  in  tlu^  other  classes.      Also,  (»f  course, 
the  prices  u.^'  and  a.!'  are  in  the  same  ratio  as  the  })rices  a.^  and 

«2  (the  prices  ordinarily  quoted),  so  that    ",,  =  —  and     ^,  =  —  , 
and  so  on.      Now  then 

V'  +  y/'-l- _  1 


V'  +  y/'+ 1 


and 


+  ]//^4- 


V^+^/'-h 


1 


(3] 


_  1 


The  interpretiition  of  these  two  equations,  along  with  w  hat  we 
have  discovered  and  presently  shall  discover  about  liie  third 
kind  of  mean  and  average,  gives  us  the  following  |»roposition  : — 
When  constant  sums  of  money  are  cxi)eirdcd  at  both  periods, 
the  true   result  is  always  obtained  (1)  by  the  luiniionic  average 


ITS    FOllMUI.ATIOX  307 

of  the  price  variations  with  weighting  according  to  tlie  nnmbers 
of  the  mass-units  equivalent  over  both  the  periods  whicli  the 
classes  contain  at  the  Jird  })eriod,  or  (2)  by  the  arithmetic  aver- 
age of  the  price  variations  with  weighting  according  to  the  num- 
bers of  such  equivalent  mass-units  which  the  classes  contain  at 
the  second  period,  or  (3)  in  cases  with  only  two  equally  impor- 
tant classes,  by  the  geometric  mean,  and  in  most  cases  (provided 
there  be  no  great  irregularity  in  the  sizes  of  the  classes)  very 
nearly  by  the  geometric  average  with  weighting  according  to  the 
geom'etric  means  of  the  numbers  of  such  equivalent  mass-units 
which  the  classes  contain  at  each  period  ^ — which  weighting  is 
the  same  as  the  weighting  of  the  classes  according  to  the  con- 
stant sizes  of  the  classes  at  both  periods.^ 

But  the  first  two  of  these  final  solutions  of  (»ur  problem  in 
regard  to  the  proper  averages  of  price  variations  (and  their 
weightings)  are  of  no  practical  utility,  since  they  presuppose 
that  we  have  performed  the  labor  of  finding  the  masses  that  are 
equivalent  over  both  the  periods  together  and  the  numbers  of 
these  contained  in  the  classes  at  one  of  the  periods.  If  we 
have  performed  this  labor,  it  would  be  easier  to  continue  the 
operation  in  the  way  jirescribed  in  the  first  of  the  above  for- 
mulae. What  we  want  is  a  method  that  will  save  us  the  great 
labor  of  such  reductions.  Here  the  geometric  mean  finds  its 
recommendation  in  the  cases  where  it  is  applicable ;  and  here 
the  geometric  average  would  be  exactly  what  we  want,  were  it 
only  accurate  in  all  cases. 

§  3.  It  is,  however,  possible  to  make  a  formula  that  will  in 
all  cases  yield  the  true  result  without  requiring  the  finding  of 
such  equivalent  mass-units — or  rather  a  modification  of  the 
above  formulse  such  as  to  contain  within  itself,  so  to  speak,  the 
method  of  selecting  the  mass-units, — a  formula  therefore,  that 

^  These  general  statements  account  for  the  particular  statements  made  in 
Notes  4,  5,  and  7  in  Sect.  I.  of  this  Chapter. 

^  For  x^"  =  — ^  and   so  on,  and   if  the  condition   is  given   that  V a^'a^' ^ 
til 

a^  b 

V  ,3i"/V'= ,   it  follows  that  l/a-i'V=  v/<V''    v'vTW'  =  V Wi'W  = 

b 


V  ai"a2",  and  so  on  ;  wherefore  V  x^'zo'  :  v  ViVo'  ■ 


308  TllK    MKTHOl)    FOi;    CONSTANT    SUMS 

is  api)licable  to  any  mass-units  wc  happen  to  use  in  the  various 
classes,  requirins::  of  us  only  that  we  know  the  prices  of  these 
ninss-units,  and  the  nuinWcrs  ol'tliciii  purcliascd,  at  cacli  period. 
Let  us  continue  to  use  the  doubly  accented  letters,  .r,",  x.^',  «,", 
a^',  and  the  like,  for  the  numbei's  and  the  prices  of  the  mass- 
units  which  are  equivalent  over  both  the  periods  ;  and  let  us, 
as  generally,  use  the  simple  letters,  x^,  x^,  a^,  a.^,  and  the  like, 
for  the  numbers  and  the  prices  of  the  ordinary  mass-units  wliich 
we  find  employed  by  merchants  and  referred  to  in  statistical  re- 
ports.    Now  it  is  possible  to  prove  that 

—  —  a      b      c 

xya^a.,  :  yy^,^.^  :  -yrji  '  ''  ^' '•  ^' '•  :,"'•  j 

'•I        r\        /i 

and 

-^  —  —  a      b      c 

.fgv/^iS  ''  y-y?\^2  '■  'h'^r{ft  '•  ::  .77/  '  oT,  '  TTTf'-  • 

"2  12  '2 

We  may  begin  with  the  first  two  terms  on  each  side  of  these 
proportions.     As  already  explained, 

«.,        a,  ^  P2       i'i. 

we  have  —  =  — r,  fi"ti         3-  =  -^  , 


«i      «i  Pi      Px 

whence  «.,  =  — ".t  a»d  it,  = 


a 


Also  by  the  hypothesis  we  have  «/'«^"  =  /9/'^5/', 

Q  n  -J  n  f.  II „  II 

whence  <  =^^  ='"*!  K' =    ' .7r  - 

'H  P\ 

Substituting  these  vahics  in  the  preceding  expressions,  we  have 

«i  Pi 

And   substitutiuii'   tliese  last  values  in   the  first  expression  to  be 
proved,  we  have 


*,„'';,^;V'/V':'/,A-^«,"<'::  J 


'■   s/„'V"-    *    •    ^-- 


But  x.a,  =  a  aiul    II J.  =  b,  ami   as  ^),",%"  =  ^/«,"«,",  tlioy 


ITS    FORMULATION  .30i) 

may  be  eliminated.      Thus  this  ])r()|)()rti()n   is  evident.      A^ain, 
by  a  similar  derivation  from  the  known  conditions,  we  obtain 

and   substituting   these  values   in   the   second  expression  to  be 
proved,  we  have 


S  /'2  '-2         r-i 


in  which  ^r/Zg  =  a  and  //^/^'^  =  b,  and  the  remainders  of  the  ex- 
pressions in  the  first  half  eliminate  as  before,  so  that  this  pro- 
portion also  is  evident. 

The  same  operation  may  be  conducted  upon  the  classes  [A] 
and  [C]  (or  upon  [B]  and  [C] ),  and  so  on  through  the  whole 
list,  with  the  same  result.     Q.  E.  D. 

Consequently 

xya,a^  +  .yX/Va  +  ^yr{f2  +  

a;X«l«2  +  3/2^A/^2  +   ^2^?'ir2  +    

a         b         c 
a         b         c 

But  we  know  by  the  second  formula  above  given  that  the  last 
half  of  this  equation  is  the  expression  for  the  general  price 
variation.     Therefore 

Pi       ;r.y^+  ?/3^/p2  +  -2^hn  +  '  ^'^^ 

Here  we  have  the  formula  desired,  applicable  to  any  mass-units 
and  their  prices  and  quantities,  in  all  cases  when  constant  sums 
happen  to  be  spent  on  all  the  classes  at  both  periods. 

This  formula  may  be  further  metamorphosed.  It  may  be  so 
stated, 


310  TIIK    MKTHOI)    FOi:    CONS'IANI'    srM> 


a     . —      b     . .  .        c     . — 


tj. 
which  reduces  to 


P.       a     .  b     .^.        c     . 

'2  ^2 


"1    "2 


and  also  to 


a  -r  ~  ^ 

Jl  ,^        ^•*'2 ^  3/2 "'^2 /"7\ 


These  are  more  curious  than  useful,  exce])t  for  theoretical   ])ur- 
poses.     The  last  leads  on  through 


-■'.  (8) 


to 

I'l    ^^i^/.'v,  +  /^y 2/13/2 + ry V2 + 


which  we  recogni/e  as  a  form  of  Scrope's  method.  It  obviously 
is  Scrope's  method  applied  to  the  geometric  means  of  the  mass- 
quantities  in  every  class  at  each  period.  Thus  the  method  we 
have  just  elaborated  turns  out  to  be  a  new  form  of  Scrope's 
method, 

§  4.  That  this  ♦nethod  does  not  violate  any  of  our  Proposi- 
tions concerning  exchange- value  in  all  other  things  goes  almost 
without  the  need  of  notic'c  Taking  any  one  of  the  formulte 
representing  it(but  most  conveniently  iS  and  S),  we  see  plainly  that 
it  indicates  constancy  if  no  prices  vary  (Pro|)ositions  XXVII. 
and  XLIV.),  and  if  all  prices  vary  alike,  it  indicates  the  same 


rrs   Foi.'.Mii.A  rioN  :]]\ 

variation  (  Prn|)(isiti(ins  XN'II.  and  XL\'.),  no  matter  what  he 
the  weights  (provided,  of  eoiirse,  they  be  the  same  at  both 
periods)  ;  nor  can  it  indicate  constancy  if  there  is  onlv  one  price 
variation,  or  if  all  price  variations  are  in  one  direction — and  in 
the  latter  event  it  naturally  cannot  indicate  a  variation  in  tiie 
opposite  direction  (Pro])ositions  XX.  and  XXVIII.).  But  there 
is  one  Proposition  by  which  it  especially  needs  to  be  tested. 
This  is  Proposition  XXXVI.  There  is  perfect  evidence  that  if 
our  method,  apjilied  to  all  l)ut  a  few  classes,  indicates  a  variation 

P. 

(or  constancy)  in  the  level  of  ))rices  so  that  ^' =/■,  and  if  we  find 

1 

that  the  remaining  classes  have  all  varie<l  in  ])rice  individually 

so  that  the  price  of  each  at  the  second  period  is  r  times  its  price 
at  the  first  (or  even  if  they  have  collectively  done  so),  our  method 
ought  still,  with  these  classes  added,  to  indicate  the  same  vari- 
ation. Let  [C]  embrace  all  the  individuals  left  over.  Formula 
(H)  is  here  the  most  convenient.  If  our  method,  applied  to  all 
classes  except  [C],  indicates  a  variation  in  the  general  level  of 
prices  from  l^to  /■,  we  must  have 

^Jf+^Ji^ ='-(^J^^J^ )■ 

r,  r       1 

But  the  ]u-ice  of  [C]  has  varied   so  that  —  =  r,  and  —  = 

/'i  T2       '■ 

Hence,  extending  our  meth(xl  to  include  [C],  we  should  have 

+"^1 

\\hich  is  evidently  true,  because  it  reduces  td 

^J^-'Ji^ +"^"='-(^J^''J^ 

+  C  n//-. 


:]\-2  riiH  METHOD  von  coNsrANT  sums 

Here  we  are  supposing  that  z.j^  =  zj^  =  C,  in  order  that  the 
chiss  [C]  may  be  properly  subject  to  this  method.  If  things 
were  not  so — if  we  had  to  distinguish  C  into  C,  and  C.„ — the  de- 
sired equality  would  not  exist.  Now  that  Proposition  ^  does  not 
require  tlie  omitted  class  or  classes  to  V)e  of  the  same  money- 
value  at  both  the  periods.  Still  this  method  is  not  shown  to  be 
false  because  it  would  not  be  true  if  we  a])plied  it  to  conditions 
to  which  it  does  not  pretend  to  be  applicable.  Confined  to  the 
conditions  to  which  it  is  applicable,  it  stands  the  test  of  that 
Proposition  pcrf(H'tly.  '* 

§  5.  That  our  formula  (5),  upon  which  depend  the  ones  fol- 
lowing, is  correctly  derived,  may  be  negatively  shown  in  this 
way.  If  the  mass-units  have  already  been  chosen  that  are 
ecjuivalent  over  both  the  periods  together,  this  formula  reduces 
to  formula  (1),  and  so  is  correct. 

That  this  formula  and  the  ones  following  it  will  reduce  to 
formulae  expressing  the  harmonic  and  tlie  arithmetic  averages, 
and  in  some  cases  the  geometric  mean,  of  the  price  variations, 
with  the  weightings  above  described  (in  §  2),  is  proved  by  the 
operation  by  which  these  formulse  themselves  were  proved,  since 
formula  (2)  to  which  they  were  proved  to  be  equal  is  equal  to 
formula  (1). 

But  witli  more  than  two  classes,  even  if  equally  important, 
this  formula  will  not  universally  reduce  to  the  geometric  aver- 
age of  the  })ri(;e  variations. 

Some  spe(!ial  cases  in  which  the  geometric  average  of  the 
price  variations  will  regularly  agree  witli  tliis  method  and  give 
the  true  result  may  easily  be  seen  to  be  as  follows  : — (1)  if  all  the 
classes  are  equally  im|)ortant   and   if  there   are   4,   (S,    1(],   32, 

64,  such   classes;  f)r   then    the  coinmon    variation   of  two 

classes  may  be  compared  with  the  connnon  variation  of  two 
other  classes,  and  the  common  variation  of  all   these  four  with 

■*  Nor  Proposition  XXXII.,  which  comes  into  phiy  if  r  =  1. 

'' A  qualificiition,  however,  must  be  added.  .Vbove  we  used  formula  ((!).  If 
we  used  foriuulie  (1),  (2),  (5)  or  (7),  the  same  inability  to  satisfy  that  Proposition 
would  remain.  But  if,  instead,  we  used  formulae  (."i),  (4)  or  (8),  (which  contain 
nothing;  that  shows  their  confiiUMnent  to  cases  with  constant  sums),  and  treated 
the  additional  mass-(|uantities  as  in  any  of  thi'iii  prcsiTibcd,  this  mctliod  would 
carry  out  tiuit  Proposition. 


DEVIATION'    OF    THE    OEO>ri:TRI('    AVERAGE  313 

the  conmioii  variation  of  four  others,  and  so  on  ; — (2)  if  some  of 

the  eUisses  are  doubly,  qua(h"U])ly,  o(!tuply, large  aud  in  the 

preceding  schema  take  the  place  of  sets  of  2,  4,  8,  classes; 

(3)  if  beside  such  two  or  more  classes  all  the  rest  are  without 
price  variation,  or  all  have  the  same  price  variation,  aud  if  the 
price  variations  of  those  classes,  measured  a])art  by  themselves, 
show  general  constancy,  or  that  same  })rice  variation  ;  for  then 
constancy,  or  that  variation,  is  the  true  result  for  all,  and  this 
will  still  be  indicated  both  by  the  geometric  average  aud  by  this 
method,  as  just  shown,  no  matter  (in  the  case  of  the  geometric 
averaging)  what  be  the  weighting  of  these  additional  classes. 
We  shall  also  find  a  more  general  account  of  other  more  irreg- 
ular cases  in  which  the  geometric  average  may  be  exactly  true. 
We  must  turn  to  the  examination  of  the  geometric  average 
of  price  variations,  this  being  generally  so  nearly  true  as  to  de- 
serve our  attention  bv  itself. 


IV. 

§  1.  A  couple  of  illustrative  examples  may  be  examined  in 
full.  We  need  to  treat  only  of  the  next  simplest  cases.  These 
are  when  we  have  three  equally  important  classes  to  deal  with, 
two  of  which  varv  alike,  or — what  is  the  same  thing- — two  classes 
one  of  which  is  twice  as  important  as  the  other.  Here  is  room 
for  two  exam])les  according  as  it  is  the  more  or  the  less  impor- 
tant class  that  falls  in  compensation  for  the  rise  of  the  other. 
Let  us  begin  with  the  former.  In  the  usual  form  of  starting 
with  prices  at  1.00,  our  familiar  example  with  A  rising  by  50 
per  cent,  may  be  schematized  as  follows — for  the  harmonic 
price  variations  : 


I  100    XOij  1.00     200    B  C'i  1.00        •       —300 
II     66|  A  r'V.  1.50     2331  b  (<<     .8571(  ^  f )— 300 

for  the  arithmetic  price  variations  : 


100  for  [A]    200  for  [B], 
100  for  [A]    200  for  [B] ; 


I  100    A  @,  1.00     200    B  @  1.00—300 
II     66 1  A  @  1 .  50    266 1  B  @    .  75—333  \ 


100  for  [A]     200  for  [B], 
100  for  [A]     200  for  [B]; 


314  Tin-:  MiyrnoD  kok  constant  sums 

for  the  u'conictric  \n'\vv  vai'iatioiis  : 

I  lUO    A  Oi   1.00     200         IW'/    1.00  —300 

1 1     (If. ^  A  Or  1 . 50     244. <)4S  I',  0,     . 81  Ko  ( =  ^/ 1)  — 3 11 . ()1 4 

I  100  for  [A]     200  for  [B], 
I  ion  for  [A]     200  fcr  [B]. 

Tn  the  first  of  these,  in  which  tlie  harmonic  average  of  the 
price  variations  with  the  Avei<::hts  1  for  [A]  and  2  for  [B]  in- 
dicates constancy,  the  com])ensation  is  (arithmetically)  by  equal 
quantities  of  the  mass-units  of  [A]  and  of  [B]  that  are  equiva- 
lent at  the  tir.st  period — a  conijiensation  wliich,  as  before,  is 
good  at  the  first  period  only,  and  is  too  small  for  both  the 
periods  together.  In  the  second,  in  wliich  the  arithmetic  aver- 
age of  the  price  variations  with  the  same  weighting  indicates 
constancy,  the  compensation  is  by  double  gain  on  [B]  over  the 
loss  on  [A]  in  the  same  mass-units,  this  double  gain  being  con- 
formable to  the  fact  that  the  price  of  this  mass-unit  of  [B]  has 
sunk  at  tlie  second  period  to  half  that  of  the  mass-unit  of  [A],  so 
that  this  compensation  is  the  compensation  of  the  second  period 
alone, — as  may  l)e  ])erceived  al.<o  by  this  arrangement  : 

1100    A  01',  l.OO     100    B' r",  2.00— 200  I  100  for  [A]     200  for  [B], 
11    6G|  A  fTA  l.oO     133i  B'  (<>'  1.00—200  |  100  for  [A]     200  for  [B], 

in  which,  A  and  W  now  being  mass-units  that  are  equivalent  at 
tlie  second  period,  tlie  compensation  is  by  an  e(pial  nnmber  of 
such,  and  so  is  too  great  f  )r  both  the  periods  together.  Thus 
the  argument  from  comjK'Usation  bv  e(|nal  mass-quantities,  ap- 
plied to  the  harmonic  and  arithmetic  averages  in  these  more 
complex  cases,  has  the  same  rcN'crsed  errors  we  liave  seen  in  the 
simjile  cases  with  two  equalK'  iiu|iortant  classes. 

In  the  third  schema,  in  which  the  geometric  average  of  the 
price  variations  with  the  same  weigiiting  indicates  constancy, 
there  is  no  appearance  of  anything  to  reconnnend  it,  except  the 
fact  that  its  compensatory  price  and  mass-cjuantity  fall  between 
the  others.  It  has  to  be  altered  so  that  the  mass-quantities 
compared  be  brought  into  better  sluqie  for  the  com])arison.  In 
it   the  ])rice  of   B  at  the  second    period   is  the   reciprocal  of  the 


i)i;\i.\i'i(».\'   oi'  'nil-;  (;k()MI"1'I!Ic  AVKi{A(iK  315 

geometric  mean   between  the  two  ))rices  of  A,  namely  1.2247. 
The  formnlation  for  [B]  may  then  l)e  changed  into  either 


I  i;«^       B'  03  1.50 
II  162.299  Ri  (^  1.2247 
or 

I  1(V2.299  B"  Oi',  1.2247 
II  200         B"  0?..  1.00 


200  for  [B], 
200  for  [B], 

200  for  [B], 
200  for  [B]. 


Still  we  have  not  what  will  iielj)  ns.  Suppose  then,  making 
use  of  a  hint  given  in  Chapter  VIII.  Section  I.  §  6,  we  take 
half  of  each  of  these  alternative  schematizations,  as  follows  : 


1   66s      B'  (».  1.50      +    81.65  B"  Or.  1.2247 
II  81.65  B'  fa,  1.2247  +  100        B"  Or>  1.00 


200  for  [B], 
200  for  [B]. 


Here,  however,  tliere  are  two  diiferent  mass-units  of  [B],  the 
second  being  .SI  Go  of  the  first.  Therefore  this  arrangement 
does  not  in  itself  help  us.  But  we  may  combine  the  numbers 
given  for  the  mass-units,  and  form  a  new  single  mass-unit,  with 
the  following  numbers  and  j)rices  : 


I  14-S.32  B"'  @  1.34846 
II  181.65  Biii(?),  1.10102 


200  for  [B], 
200  for  [B]. 


Here  the  ])eculiarity  exists  that  the  diiference  in  the  numbers 
of  these  mass-units  is  33|^,  so  that  we  have  compensation  offered 
l)y  equal  mass-cpiantities.  Now  if  the  geometric  mean  of  the 
prices  of  this  mass-unit  were  the  same  as  the  geometric  mean  of 
the  prices  of  A,  we  should  have  exactly  what  we  want;  for 
then  this  mass-unit  of  [B]  would  have  the  same  exchange-value 
over  both  the  periods  as  the  mass-unit  of  [A] ,  and  the  gain  of 
33J  of  these  mass-units  of  [B]  would  exactly  equal  and  com- 
pensate the  loss  of  33 J  of  the  mass-units  of  [A].  Unfortu- 
nately this  is  not  the  case.  By  the  nature  of  the  formation  of 
this  mass-unit  and  its  prices,  its  prices  have  1.2247  for  their 
arithmetic  mean  only,  and  their  geometric  mean  is  a  trifle 
lower,  being  in  effect  1.2185.  Thus  these  prices  of  B'^^  incline 
in  the  same  direction  as  the  prices  in  the  harmonic  extremes, 
also  offering  compensation  by  e(i[ual  mass-quantities  ;  for  the 
mean  between  those  prices  of  B  is  far  below  that  between  the 
prices   of  A.      We   know  that  in  those  variations  the  price  of 


31()  THK    METHOD    FOR    CONSTANT    SUMS 

[V)~\  has  not  fallen  far  enough.  So,  too,  here  we  have  an  indi- 
cation that  the  priee  of  [B]  has  not  fallen  quite  far  enough. 
There  is  compensation  by  gain  of  an  e<|ual  number  of  mass-units 
sliii'htlv  less  valuable  than  the  mass-units  lost.      Therefore  monev 

o  * 

has  slightly  depreciated,  and  the  })rice-level  has  risen  somewhat. 
Another  hint  offered  in  the  same  place  in  Chapter  VIII. 
leads  to  the  same  conclusion.  There  we  saw  that  it  is  possible 
to  arrange  things  so  that  the  prices  of  two  similar  mass-units  of 
[B] ,  falling  as  from  1.00  to  .SI 05,  together  traverse  a  distance 
equal  to  that  traversed  by  one  A,  namely  when  they  fall  from 
l.:5G2:i  to  1.1123.     The  schema  for  [B]  would  then  be 

I  146.81  Biv  (if,  1.362.3  I  200  for  [B], 
II  179.80  Biv  @  1.1123  I  200  for  [B]. 

Here  the  difference  in  the  quantities  is  a  gain  of  32.99  such 
mass-units.  The  geometric  mean  of  these  prices  is  1.23096. 
Thus  while  the  compensatory  quantities  are  slightly  less,  the  ex- 
change-value of  their  units  over  both  the  periods  is  slightly 
greater.  Which  of  these  differences  preponderates?  On  [A] 
we  lose  1.22474  x  33|^  =  40.824  money-unit's  worths  (over 
both  the  periods  together);  on  [B]  we  gain  1.23096  x  32.99 
=  40.609  similar  money-unit's  worths.  So  we  gain  less  than 
we  lose,  and,  as  before,  our  money  has  depreciated,  the  fall  in 
the  price  of  [B]  not  being  quite  great  enough. 

Now  it  is  easy  to  place  the  prices  of  a  mass-unit  of  [B] ,  fall- 
ing as  from  1.00  to  .<S165,  around  1.2247  as  their  geometric  mean, 
and  to  find  the  numbers  of  these  mass-units  purchasable  with 
200  money -units  at  each  period.     They  are 


I  147.5579  B''  dj)  1.3554 
II  180.7174  B'^@  1.10(i7 


200  for  [I!], 
200  for  [B]. 


But  here  the  gain  in  such  mass-units  is  33.1595,  or  a  trifle  too 
little.  This  confirms  the  previous  conclusion  that  the  price  of 
[]>]  has  not  fallen  (juite  far  enough  ;  for  it  indicates  a  slight 
dimiiuition  in  the  purchasing  power  of  oui-  money,  since  we  are 
not  ])ermitted  to  gain  (piite  so  many  (Kjiially  Nahiable  mass-units 
of  [B]  as  we  lose  of  [A]. 

What  we  want  is  to  combine  certain  features  which  have  so 


DEYIATIOX    OF    THE    (JEOMETKIC    AVEI{A(tE  'Ml 

far  not  come  togotlier,  and  which  will  not  c<Mne  together  so  long 
as  we  suppose  the  price  of  [B]  to  fall  as  from  1.00  to  .8165. 
We  want  a  mass-unit  such  that  its  prices,  falling,  have  1.2247 
for  their  geometric  mean,  and  such  that  the  numbers  of  them 
purchasable  with  200  money-units  at  the  first  and  at  the  second 
period  increase  by  33J.  If  these  two  objects  were  obtainable 
with  a  fall  of  the  price  of  [B]  as  from  1.00  to  .8165,  the  geo- 
metric average  of  price  variations  would  be  true  in  this  and  all 
such  cases.  But  they  are  not,  and  we  must  seek  for  another  fall 
of  the  price  of  [B]  that  will  unite  these  requirements.  It  is 
not  difficult  to  find  the  terms  of  this  desired  fall.  The  follow- 
ing is  the  schema  for  [B]  : 


I  147.480987  B''  («  1.856107 
II  180.814333  B''  @  1.106107 


200  for  [B], 
200  for  [Bj. 


Here  we  have  a  class,  [B] ,  twice  as  large  as  the  class  [A] ,  and 
the  price  variations  are  such  that  for  what  we  lose  ou  the  mass- 
quantity  of  [A] ,  the  price  of  whose  unit  varies  around  1.2247  as 
its  geometric  mean,  we  gain  exactly  as  much  on  the  mass-quantity 
of  [B],  the  price  of  whose  unit  varies  around  the  same  figure, 
1.2247,  also  as  its  geometric  mean,  and  whose  units  gained  are 
equal  in  number  to  the  other  units  lost.'  Therefore  our  gain  is 
exactly  equivalent  to  our  loss  over  both  the  periods  compared. 
Therefore  our  money  has  retained  exactly  the  same  purchasing 
power  or  exchange-value.      But  what  is  this  ])rice  variation  ? 

1-106107  ,  ,.  ,    , 

It   IS   as  from  1.00  to  —^-—^- =  .815648 — a   figure    slightly 

lower,  as  we  have  already  found  reason  to  expect,  than  the  fig- 
ure required  by  the  geometric  method.  Therefore  if  the  price 
of  B  falls  only  from  1.00  to  .8165,  at  which  point  the  geometric 
average  indicates  constancy,  it  has  not  fallen  quite  far  enough  to 
counterbalance  the  rise  of  A  from  1.00  to  1.50,  so  that  the  gen- 
eral level  of  prices  has  slightly  risen,  and  the  geometric  average 

1  It  may  be  noticed  that  the  geometric  mean  between  the  numbers  of  these 
mass-units  of  [B]  is  10-3.3,  wliicli  is  just  double  tlie  geometric  mean  between  the 
numbers  of  the  mass-units  of  [A],  namely  81.6.5.  Tliis  is  in  argument  with  Note 
2  above  in  Sect.  II.  It  should  also  be  noticed  that  the  prices  of  every  two  such 
mass-units  of  [B]  together  traverse  in  the  opposite  direction  exactly  the  same  dis- 
tance as  the  price  of  the  one  mass-unit  of  [A]. 


318  'iiii;  .Mi:rin»i'   I'oi;  cons-ian'I"  sims 

has,  ill  tliis  case,  iikkIc   an    indication    slightly    hclow   the  truth. 

Wo  slidiild  ha\'('  i)('cn  pleased  if"  one  of  the  faniilai'  and  easil\- 
working"  avera<;os,  a[)[)lie(l  to  the  |)riee  variatiitns  with  weight- 
ing according  to  the  constant  snnis  ex|)en(h'd,  in  this  case  1  for 
[A]  and  '1  for  [B],  would  indicate  constancy  only  when  the 
price  of  [B]  has  fallen  from  l.(H)  to  .Sl.")()4s.  Unfortunately 
noue  of  the  three  do  so."  J^ut  of  the  three  the  geometric  aver- 
age, with  this  weighting,  comes  the  nearest  to  it. 

Ill  defanlt  of  a  true  average  with  the  oidy  weighting  conve- 
niently indicated,  we  can,  of  course,  resort  to  one  of  the  formulae 
above  discovered,  preferably  (5),  and  directly  find  the  correct 
result  without  going  through  the  long  operation  above  pursued, 
which  has  been  done  to  illustrate  the  princi|)le  and  to  show  the 
defect  in  the  geometric  average.  Thus  applied  for  the  jmrpose 
of  finding  the  proper  variation  of  the  ))rice  of  B  from  1.00  when 
A  rises  from  1.00  to  1.50  and  at  both  jx-i-iods  100  money-units 
are  spent  on  [A]  and  '2(H)  on  [P>],  in  order  to  have  constancy, 
the  formula  is 

100  X  1.224744  -}-  200n/,I^ 

200     - 
im  X  1.224744 -h     .    v/;t 

This  reduces  to  ,9^  -|-  0.204124%/ 9.,  =  1  .oo,  which  being  worked 

(»ut  in  the  ordinary  algel)raical  manner  (with  only  positive  terms) 

gives  v/^2  =  .903132,  and  ,i=  .,Sir)(;4S. 

§  2.   The  second  example,  in  which  the  class  with  rising  j)rice 

is  the  more  important,  may  be  disposed  of  more  l)riefiy.      Let 

us   su|)pose  that   twice  as  much    monev  is  spent   on    [A]   at  each 

-  Or  we  sliould  be  almost  as  well  pleased  if  we  hail  some  easily  indicated  sys- 
tem of  weighting  which  with  one  of  the  familiar  averages  would  give  the  true 
result.  Now  in  the  above  example,  when  V>  falls  from  1.00  to  .81.'>(i4^;,  the  weight- 
ing wanted  with  the  geometric  average  is  1  for  [A]  and  1.9.s!t7  for  [B] ;  with  the 
harmonic,  1  for  [A]  and  1.4748  for  [B] ;  with  the  arithmetic,  1  for  [A]  and  2.7122 
for  [B].  Here  the  weighting  for  the  geometric  average  is  hopeless.  That  for  the 
harmonic  average  is  the  ratioof  the  number  of  the  (over  both  periods)  equivalent 
mass-units  of  [,\]  to  those  of  [B]  purchased  at  the  first  period  ;  and  that  for  the 
arithmetic  average  the  ratio  between  them  at  the  second  period.  This  is  in  agree- 
ment with  wiiat  is  above  shown  in  Sect.  II.  {^2,  which  gives  us  a  general  principle 
for  the  wcigliting  of  these  averages,  l)ut  one,  as  ihere  remarked,  ton  lal)i)ri()us  to 
))e  of  utility. 


l)i:\'IATI()N    OF    THE    (iKOMKTItK '    AVEIJA(;E 


311) 


period  as  on  [B].  Now  tor  (■(•iistaiicv  to  l)e  indicated  l)y  the 
three  kinds  of  averages  each  witii  the  weights  2  for  [A]  and  1 
for  [B],  the  s(4iemata  must  be  as  follows — for  the  harmonic 
averasre  : 


I  200    A  @  1.00     100    B  @  1.00- 
II133iA@1.50     166fB@    .60- 

for  the  arithmetic  average  : 


-300 
-300 


I  200    A  @  1.00     100  B  @,  1.00—300 
II  133^  A  @  1.50      00   B@     0 

for  the  geometric  average  : 


200  for  [A]      100  for  [B], 
200  for  [A]     100  for  [B]; 


200  for  [A]     100  for  [B], 
200  for  [A]     ; 


I  200    A  @,  1.00     100  B  @  1.00  —300 

II  133i  A  r^>,  1.50     225  B  @    .4444  ( =  ^ )— 358^ 


200  for  [A]     100  for  [B], 
200  for  [A]     100  for  [B]. 


In  the  first  the  compensation  is  still  arithmetically  by  equal 
mass-quantities,  measured  in  equivalents  at  the  first  period,  and 
therefore  only  the  proper  compensation  at  the  first  period  itself. 
In  the  second  the  required  change  of  price  of  B  is  to  zero,  which 
makes  A  at  the  second  period  infinitely  more  valualile  than  B  ; 
wherefore  com})ensation  is  impossible.  The  third  again  needs 
to  be  restated.  The  restatement  may  be  made,  this  time,  by 
changing  the  mass-units  of  [A]  .  The  price  of  A  at  the  second 
period,  1.50,  is  the  reciprocal  of  the  geometric  mean  between 
the  two  })rices  of  B.      We  may  then  have 


I  225  Ai  (Tr^  .4444  +  150  A"  @    .6666 
II  150  At  (<>:.  .6666  -1-  100  A"  (q>j  1.00 


200  for  [A], 
200  for  [A]; 


whence,  by  combining  the  quantities,  and  forming  a  new  mass 

unit  : 

I  375  A"'  @  .5333  (=  /j)  [  200  for  [Al, 
II  2.iO  A'"  (o)  .80  I  200  for  [A]; 

Here  the  number  of  these  mass-units  lost  is  the  same  as  of  the 
mass-units  of  [B]  gained.  But  the  prices  are  arithmetical, 
not  geometrical,  terms  around  .666(3,  and  their  geometrical 
mean  is  lower,  being  .6532.  The  conditions  no  longer  resem- 
ble those  in  the  harmonic  schema,  in  which  the  mean  of  the 
prices  <^f  the  mass-unit  which  in  equal  numbers  compensates  for 
the  gain  of  mass-units  of  [B]  is  above  the  mean  of  the  prices 


320  THE    METHOD    lOi;    CONSTANT    SUMS 

of  the  lattei".      The  suggestion   ihnu   this  ditferenee   is  that   the 

price  of  A  has  not  risen  (juite  high  cnoiigli.      We  lose  an  (([iial 

number  of  mass-units  that  are  not  quite  so  vahiable  (over  both 

the  periods)  as  the  mass-units  gained.     Therefore  our  money 

has  augmented  in  purchasing  power,  and  the   price-level  has 

fallen  somewhat.     The  same  suggestion  is  made  when  we  liiid 

the  mass-unit  of   [A]  whose  j)rices,  rising  in  the  same  })ropor- 

tion  as  before,  do  vary  around  .6*) (5 6  as  their  geometrical  mean, 

as  follows  : 

I  367.42349  A''  f?,  .544331  j  200  for  [A], 
II  244.95007  .V  On  .81G497  !  200  for  [A], 

the  numl)er  of  these  mass-units  lost  being  smaller  than  that  of 
the  equally  (over  both  the  periods)  valuable  mass-units  of  [B] 
gained,  so  that  money  has  slightly  appreciated.  To  get  the 
proper  diiference,  125,  of  mass-units  of  [A]  whose  prices  are  in 
the  same  variation  around  .6666  as  their  geometric  mean,  we 
find  the  following  conditions  required  : 

T  3()8.942  A'^  @  .54209  j  200  for  [A], 
II  243.941  A'^  @  .81987  \  200  for  [A]- 

Here  at  last  we  have  the  proper  compensation,  and  the  pr()})er 
price  variation  of  the  class    [A]  .^     The  proper  price  variation 

is  a  rise  as  from  l.OO  to    ^-,^777,  =  l.')124.     The  same  fioure  is 

.o42()l' 

obtained  more  directly  by  use    of  the  above-discovered  fbrnuda 

(5).      Ap})lied   to  mass-units  priced  at   1.00  at  the  first   period, 

this  becomes 


200  s/a,  +  100  X  t 


=  1 


which  reduces  to  «.,  —  0,41()()6()\/«.,  =  1,  and  works  out,  giving 

x/^",  =  1.229804   and   a.,=  1.5124.     The  figure  is,  this  time, 

as  we  have  found  reason  to  expect,  slightly  higher  than  required 

by  the  geometric  average.      The  geometric  average  is  satisfied, 

^The  geometric  iiicaii  hctwceii  tlic  iiuinbers  of  these  mass-units  of  [A]  is 
double  the  geometric  iiu-aii  l)etween  the  numbers  of  the  mass-units  of  [li];  and 
the  prices  of  every  two  such  mass-units  of  [A]  togetiier  traverse  the  same  distance 
as  the  one  mass-unit  of  [R].     Cf.  above,  Note  1. 


DKVIATION    OF    THE    GEOMETRIC    AVEI{A(;E  321 

and  indicates  constancy,  wIkmi  the  price  of  A  has  i-iscn  only  to 
1.50,  or  before  it  has  risen  to  its  proper  height,  that  is,  when  the 
compeusatiou  oifered  bv  its  variation  is  not  yet  complete,  and 
the  general  level  of  ])rices  lias  fallen.  Thus  in  this  case  the 
geometric  average  yields  a  result  slightly  above  the  truth.'' 

§  3.  These  examples,  in  conjunction  with  many  others  of  a 
similar  nature  to  be  given  later,  permit  us  to  make  an  inducti(m 
and  to  advance  the  following  rule  : — W J  ten  the  prices  that  full 
below  the  general  average  are  of  prepotidcrating  olat^ses,  the  geo- 
metric average,  with  weighting  according  to  the  constant  .vu//(.s'  of 
moneif  spent,  i/ields  a  refill  beloic  the  truth;  and  irhen  the  prices 
that  rise  above  the  general  average  are  of  pre fjond crating  classes, 
it  yields  a  result  above  the  truth. 

We  may  form  some  idea  of  the  probable  amount  of  error  in- 
curred by  the  geometric  average,  by  measuring  the  errors  in  the 
above  examples.  In  the  first,  the  sums  of  the  (over  both 
periods)  equivalent  mass-units  of  [A]  and  [B]  purchasable 
with  100  and  200  money-units  at  the  first  and  second  periods 
were  respectively   247.5571)   and  247.3840  ;    by  dividing   the 

P 

former  by  the  latter  we  get  ^^  =  1 .00<^7.       Thus  the  geometric 

1 
average,  Aveighted  according  to  the  constant  sums,  indicated  con- 
stancy where  there  really  was  a  rise  of  0.07  per  cent.  In  the 
revision  we  saw  that  there  would  really  be  constancy  if  the  price 
of  B  fell  to  .81564.  If  it  did  so,  the  geometric  average  would 
be  0.99932,  indicating  a  fall  of  0.068  per  cent.  In  the  second 
example,  the  sums  of  the  (over  both  periods)  equivalent  mass- 
units  of  [A]  and  [B]  purchasable  with  200  and  with  100 
money-units  were  successively  467. 42349  and  4() 9. 9 500 7,  whence 

P 

— ^  =  0.9946,  indicating  a  fall  of  0.54  per  cent.,  although  the 

geometric  average,  weighted  as  above  described,  indicated  con- 
stancy.     In  the  revision  we  found  that  to  obtain  constancy  the 

*  In  the  corrected  example,  with  A  rising  from  1.00  to  1.5124,  to  indiciite  con- 
stancy, the  weight  wanted  for  [A],  that  for  [B]  being  1,  with  the  geometric  aver- 
age is  1.9602,  with  the  harmonic  3.G89,  with  the  arithmetic  1.0S4— the  latter  two 
being  the  ratios  between  the  numbers  of  the  equivalent  mass-units  at  the  first  and 
at  the  second  periods  respectively.     Cf.  above,  Note  2. 

21 


322  'riiK   MiriiioD   I'oi;  (onstani'  >rM> 

price  of  A  should  ri.-c  t«>  l.-")l"_M.  li'  it  did  so,  tlic  ucoinct  I'ic 
average  would  We  l.OO.")-"),  indicatinu  a  ims<-  of  ().")")  ])('r  cent. 

These  trifling  errors,  wc  may  notice,  arc  on  larger  price  \-ari- 
ations  than  ordinarily  occur  in  practice  hctwecn  two  successive 
periods.  It  is  oulv  with  \-erv  large  price  variations  that  the 
deviations  become  a])|)recial)le.  But  also  iucipiality  in  the  sizes 
of  the  classes  has  something  to  do  with  the  matter.  If  the  j)re- 
ponderating  class  varies  nuich  in  price,  while  the  smaller  class 
varies  little,  the  deviation  may  he  apprcciabl(%  if  the  preponder- 
ance is  moderate,  hut  may  he  diminished  almost  to  nothing  if 
the  preponderance  is  excessive  (the  double  excess  having  the 
eifect  of  dragging  both  the  measurements  after  it  till  they  almost 
coincide) — while,  of  course,  a  very  slight  |)i'eponderance  also 
unites  the  results,  approaching  toward  evenness  of  weighting. 
On  the  other  hand,  if  the  preponderating  class  \aries  little  and 
the  smaller  class  varies  excessively,' the  deviation  may  be  great, 
and  the  g-eometric  method  becomes  wholly  untrustworth\-. 

Examples  of  these  rules  may  be  given  at  landom  as  ibllows  : 

r  10  A  @  1.00       40  B  0<   1.00  I  10  for  [A]     40  for  [I!]. 
II     5  A  @  2.00     400  lU"     .10  I   10  for  [A]     40  for  [H]. 

The  proper  result  is  p'  =  (>.2()()')  ;  the  geometric  method  yields 
0.1821. 

I  10  A  ((>}  1.00         40  P.  Oi   1.00  I   10  for  [A]     40  for  [15], 
II     o  A  @  2.00     4000  I U"     .01   I   10  for  [A]      10  f..r  [!'.]. 

l\ 

The  proi)er  result  is  p"  =  ().(>44."')  ;   the  geometric  uiethod  yields 

0.()2S.S. 

I   10  \  ('I:  1.00         400  IJ  (<l   1.00  I  10  for  [.\]     400  for  [B], 
II     7)  A  (,jj  2.00     40000  B  {«,    .01  |  10  for  [.V]     400  for  [B]. 

1>, 

The  i)ro]>er  residt  is     "=  0.0  1  14  ;   the  geometric  method   yields 

I 
0.0110. 

■'  Remember  that  falling  variations  are  larger  than  rising  variations  tiie  ner- 
ceiitages  of  which,  reckoned  in  tiie  usual  way,  arec(iual.  Anil  this  superiority 
rapidly  increases  with  the  amount  of  the  variation.  A  iall  from  1.00  to  0.!»!l  is 
only  slightly  greater  than  a  rise  from  I.OOtol.Ol.  A  r;ill  from  1.0(»  to  O.-'iO  is  twi<-e 
as  great  as  a  rise  from  l.(Mni>  1  ..")n.  A  fail  from  I  .no  to  (t.oi  is  one  hnndri'il  (imes 
as  great  as  a  rise  from  1.(M»  Io  \.W. 


DKVIATIOX    OF     rilK    ( i  i;<  )METRIC    AVKUAGK  o'io 

I    1(10  A  ('I    1.(10         10  H  C'7)  1.00  I  100  for  [A]      10  for  [H], 
II     50  A  r^A  2.00     1000  B@    .01  |  100  for  [A]     10  for  [K]. 

P., 

The  proper  result  i.Sp"=  U.(S.'}42  ;  the  geometric  method  yields 

1.2.355. 

Among  many  classes,  if  some  large  ones  rise  in  price  while 
other  large  ones  fall,  and  several  small  ones  similarly  offset  one 
another,  and  if  the  balances  left  over  of  large  and  small  classes 
that  vary  in  the  same  direction  are  small  compared  with  the 
whole,  the  source  of  error  again  is  small. 

For  the  errors  are  not  cumulative.  That  is  to  say,  if  the 
class  [A]  is  twice  as  large  as  the  class  [B] ,  and  the  class  [C] 
twice  as  large  as  the  class  [D] ,  and  the  prices  of  both  [A]  and 
[C]  are  the  rising  ones,  so  that  in  each  comparison  of  [A]  and 
[B]  and  of  [C]  and  [D]  the  error  may  be,  say,  about  half  of 
one  per  cent.,  the  error  will  be  no  greater  in  the  comparison  of 
all  four  together.  On  the  contrary,  the  error  will  be  an  aver- 
age somewhere  between  all  the  errors  in  the  comparisons  of  any 
two  unevenly  matched  pairs.  And  so,  instead  of  accumulating, 
the  errors  tend  to  neutralize  one  another.  This  follows  by  the 
principle  of  continuity  from  the  two  halves  of  the  statement 
above  advanced.  For  if  in  one  pair  the  preponderating  class  is 
the  rising  one,  the  error  is  above  the  truth ;  and  if  in  another 
pair  the  preponderating  class  is  the  falling  one,  the  error  is  be- 
low the  truth.  These  two  pairs  of  variations  taking  place  to- 
gether, and  being  averaged  together,  the  error  must  lie  between 
them,  and  so  must  be  diminished.  It  may  perhaps  disappear 
altogether. 

There  is  also  another  way  in  which  there  may  be  neutraliza- 
tion. This  is  through  a  series  of  successive  periods.  If  during 
one  comparison  of  two  periods  a  large  class  rises  in  ])rice 
through  abnormal  causes,  its  variation  will  tend  to  cause  error 
in  the  geometric  average  above  the  truth.  If  then  in  the  com- 
parison of  the  next  or  some  soon  succeeding  period  its  price  falls 
back  to  its  usual  state,  this  reverse  variation  will  tend  to  cause 
error  in  the  geometric  average  below  the  truth.  Hence  in  the 
series  the  later  error  will  make  up  for  the  earlier.      That  this 


;3'24  IIIi;    -MKTJIOI)    FOR    ( O.NS'IANT    SUMS 

kind  of  compensation,  if  tlie  reverse  cliange  is  back  to  the  same 
figure,  the  classes  remaining  of  the  same  importance,  is  exact, 
will  be  slidwn  presently. 

In  the  harnionie  and  arithmetic  averages  there  is  no  such 
tendency  toward  neutralization  in  successive  periods.  Their 
errors  may  accumulate  indefinitely. 

V. 

^  1.  What  has  been  proved  in  the  preceding  Sections  may  be 
confirmed  on  a  new  line  of  reasoning,  which  also  yields  instruc- 
tion on  points  not  yet  ])rove(l. 

Su|)|>ose  that  in  our  simple  hypothesis  of  a  world  with  money 
and  two  equally  important  commodities  we  have  the  following 
conditions,  (the  mass-units  being  chosen  in  the  usual  way,  as 
equivalent  at  the  first  period),  over  three  successive  periods  : 


I  100    A  @,  1.00     100    B  (,<>,  1.00 

II    66tA@1.50     iSSiB(a),    .75 

III  100    A  @  1.00     100    B  @  1.00 


100  for  [A]  100  for  [B], 
100  for  [A]  100  for  [B], 
100  for  [A]     100  for  [B]. 


Here  we  have  two  sets  of  price  variations,  the  first  consisting  of 
these,  I  and  f,  the  second  of  these,  |  and  |-.  In  addition  we 
have  the  comparison  between  the  prices  at  the  first  period  and 
the  j)rices  at  tlie  third  period.  This  comparison  shows  con- 
stancy ;  for  evidently,  the  state  of  things  being  the  sauie  at  the 
third  as  at  the  first  period,  we  have  the  same  (;onditions  as  if 
the  state  of  things  at  the  first  period  had  remained  constant. 
We  know,  therefi>re,  that  after  drawing  the  average  of  the  first 
set  of  price  variations  and  tliat  of  the  second  set,  if  we  then 
atteuipt  from  these  averages  to  Hud  the  average  of  the  whoh' 
variation  (oi-  constancy)  from  the  fii'st  to  the  third  ])eriod — whicii 
is  done  by  nudti])lying  together  the  two  singh'  averages  of  the 
price  variati(ms, — the  result  siiould  be  unity,  indicative  of  con- 
stancy. Here  we  have  control  oxer  the  result,  and  may  use 
these  conditions  as  a  test  (?asc. 

It  may  be  noticed  at  tiu-  outset  that  in  all  such  cases  (con- 
fined to  three  })cri()ds)  the  formuhe  above  obtained  satisfy  this 
test.      For  the  (|uantities  and   ])rices  being  exactly  the  same  at 


TK8T    CASES  325 

the  tliird  as  at  the  first  period,  th(>  first  variation  is  indicated, 
bv  fornnda  (o),  to  be 

P,  ^  '^,  v/«,^  +  .yyM  + 

I',    .v.yo.^a,  +  //y,^„T,  +  ' 

and  tlie  second 

1",    .i;y«,«^  + //y,V2  + ' 

p.,/    p.,  p. \ 

These  are  reciprocals  ;   wherefore  ^'  I  =^  ^' .  ^'  |  =  1.00.     It  is 

plain  that  if  the  |)rices  are  Irj-cf/ti/ar/i/  different  at  the  third 
period,  the  direct  comparison  of  the  third  with  the  first  will  not 
yield  the  same  result  as  the  indirect  comparison  of  them  through 
the  intervening-  period.  We  shall  not  pause  to  consider  this  in- 
consistency here,  as  it  will  again  call  for  attention  in  the  com- 
plete method  involving  this  partial  method.  We  shall  at  pres- 
ent devote  our  attention  to  the  certain  test  yielded  by  a  revision 
of  everything  at  a  later  period  to  what  it  was  at  the  first.  Let 
us  now  see  Iiom'  the  averages  of  the  [)rice  variations  stand 
this  test. 

Averaging  the  al)ove  sets  of  price  variations  harmonically,  we 

•2                                                            2  8 

get,  for  the  first,  ^       ^  =  1.00,  and   for  the  second,  ^ g  =  ^ 

t  +   3  2+9 

=  0.88<S8  ;  and  the  j)roduct  of  these  results  indicates  that  be- 
tween the  first  and  the  third  period  the  general  level  of  prices 
has  fallen  by  11.11  per  cent.,  which  we  know  to  be  wrong — and 
to  err  by  placing  the  level  of  prices  too  low. 

But  suppose,  after  drawing  the  harmonic  average  of  the  first 
set,  we  draw  the  arithmetic  average  of  the  second,  likewise  with 
even  weighting.  We  then  have  I  (|  +  |^)  =  1.00  ;  and  each 
of  these  results  indicating  constancy,  the  whole  result  indicates 
constancy,  which  is  right. 

In  effect,  the  second  price  variations,  from  the  harmonic  terms 
on  opposite  sides  of  unity  to  unity,  are  (simple)  arithmetic  vari- 
ations— that  is,  are  the  same  as  variations  from  unity  to  the 
opposite  arithmetic  terms.      Thus  at  the  second  period  the  price 


326  Till-;   Mi;rii(»i>   i-(»i;  ( onstani-  sims 

of  §  A  i.<  l.OO  and  the  |)ricc  ol"  1  .^  B  i>  !.(»().  and  at  the  third 
the  prices  of  these  parts  of  A  and  H  are  .({(ijr  and  l.-">.")^  re- 
spectively, whieh  are  the  arithnietie  terms  around  unity.  Nat- 
urally, therefore,  only  the  arithmetic  averajxP,  with  even  weight- 
ing, will  indicate  constancy  when  prices  vary  in  this  way. 

Turning  our  attention  to  the  (piantities  purchased,  we  see 
that  with  everv  two  money-units  we  purchase  at  the  first  and 
third  periods  1  A  aud  1  B,  and  at  the  second  |  A  and  ^  B. 
Thus  at  the  second  period,  compared  with  the  first,  on  the  quan- 
tities of  the  first,  the  loss  on  A  is  one  tliird  and  tlie  gain  on  B 
one  third,  the  gain  equalling  the  loss.  But  at  tlie  tliird  period, 
compared  with  the  second,  and  with  the  gain  and  loss  measured 
in  the  same  way,  that  is,  on  the  quantities  at  the  earlier  of  the 
two  periods  compared  as  the  unit  wholes,  the  gain  on  ^  A  is  one 
half  and  the  loss  on  |-  B  is  one  (juarter,  constituting  the  com- 
pensation by  harmonic  proportions.  This  agrees  with  the  fact 
that  the  price  variations  in  this  case  are  arithmetic  variations. 
But  if  the  price  variations  really  had  been  harmonic;  variations — 
namely,  a  fall  of  the  price  of  f  A  from  1.00  to  .()6f  (bringing 
the  price  of  A  back  to  1.00)  and  a  rise  of  the  price  of  |  B  from 
.33J  to  .GG|  (carrying  the  jn-ice  of  B  up  to  1.50),  so  that  the 
second  harmonic  average  would  indicate  constancy,  then  not  only 
would  the  level  of  prices  at  the  third  period  be  openly  higher 
than  at  the  first,  but  it  is  plain  that  the  compensation  offered 
would  be  a  gain  on  [A] ,  now  the  article  falling  in  price,  smaller 
than  the  loss  on  [B],  now  the  rising  article,  so  that,  losing 
equally  on  [A]  in  the  first  variation,  but  gaining  less  in  the 
second,  we  slioidd  lose  on  the  whole. 

§  2,   Again  if  we  had  these  conditions  : 


1  100    A  ("}  1.00     100  B@  1.00 

II      m'fi  A  On  l..-)0     200  B  Oi]>    .50 

III    100     A  Or  1.00     100  nOi    1.00 


100  for  [A]  100  for  [B], 
100  for  [A]  100  for  [B], 
100  for  [A]      100  for  [B], 


and  averaged  the  two  sets  of  price  variations  arithmetically,  with 
even  weighting,  the  result  for  the  first  set  windd  be  1.00,  indi- 
cating constancy,  and  that  for  the  second  would  be  1.33^-  indi- 
cating a  rise  of  33^  per  cent.      But  if  we  averaged  the  second 


TKST    (ASKS  .VI  ( 

set  liaiMiKiiiicalK',  with  I'vcn  wciii'litiiia-,  we  slumld  have  1.00, 
aiul  therefore  1.00  for  the  whole  variation,  correctly  iiulicatinti; 
constancy.  In  effect,  the  second  set  of  j)rice  variations  are  har- 
monic variations.  If  they  really  were  arithmetic,  such  that  the 
arithmetic  average  would  indicate  constancy,  the  price-level 
would  really  have  fallen  at  the  third  period  ;  for  though  the 
pri(r  of  A  w'oidd  be  back  at  1.00,  the  price  of  B  would  be  .0<)|  ; 
and  our  money,  still  spent  evenly,  would  purchase  more  than 
before. 

§  '.].  The  order  in  which  the  two  averages  haA^c  been  alter- 
iKitely  applied  might  be  inverted,  and  yet,  although  the  particu- 
lar results  would  be  different  in  every  instance,  yet  the  whole 
results  would  be  the  same.  Thus  in  the  first  example  the  arith- 
metic average  of  the  first  set  of  price  variations  would  indicate 
a  rise  of  12|^  per  cent.,  and  the  harmonic  average  of  the  second 
set  would  indicate  a  fall  of  11.11  per  cent.;  but  a  fall  from  1.12") 
by  11.11  per  cent,  is  a  fall  to  1.00.  And  in  the  last  example 
the  harmonic  average  of  the  first  set  of  price  variations  would 
indicate  a  fall  of  25  per  cent.,  and  the  arithmetic  average  of  the 
second  set  would  indicate  a  rise  of  83 J  per  cent.;  but  the  net 
result  of  the  two  variations,  |^  x  4=1,  indicates  constancy. 

From  these  particular  exam])les  we  get  the  following  general 
rules,  the  universality  of  which  is  proved  in  Appendix  B  1\. 
§§  4,  5,  7,  that  when  the  prices  of  two  equal/ 1/  hnportani  cldaxcx 
after  varying  at  tlie  second  period  revert  at  tJie  tJiij-d  to  irhat  thcif 
were  before,  the  continuous  use  of  the  harmonic  averac/e  of  the 
price  variations,  ivith  even  weighting,  gives  a  fiiud  result  known  to 
be  too  low,  and  the  continuous  use  of  the  arithmetic  average  of  the 
price  varicdion>^,  with  even  weighting,  gives  a  final  result  known  to 
he  too  high  ;  hut  the  alternate  use  of  these  two  averages,  similarli/ 
weighted,  gives  the  final  result  known  to  he  right.  We  have  suffi- 
cient acquaintance  wath  these  averages  to  infer  that  these  state- 
ments will  extend  to  include  all  cases  with  any  number  of  classes 
of  any  sizes,  the  only  provisos  being  that  constant  sums  be  ex- 
])ended  on  the  classes  at  all  the  periods,  and  that  the  weighting 
be  according  to  them. 

It  must  not  be  inferred,  hoAvever,  that  an  alternate  use  of  the 


328  iMiK  Nnrrnoi)   I'oi;  constant  sims 

arithmetic  and  harinouic  averages  wniild  at  the  end  of  every 
other  period  give  the  right  result.  Unless  all  the  prices  revert 
to  what  they  were  at  the  start,  and  the  weighting  is  the  same 
throughout,  the  correction  oifered  by  tlu;  error  of  the  one  aver- 
ao'e  for  the  error  of  the  other  has  not  full  opportunity,  or  has 
too  much  opportunity,  to  work  itself  out.  Yet  this  use  of  them 
would  de])art  less  from  the  truth  than  the  continuous  use  of 
either  of  them  alone. 

The  inverse  nature  of  these  two  averages  shows  the  mistake 
of  a  procedure  whi(^h  has  not  infrecpiently  been  pursued.  Sev- 
eral statisticians,  taking  souk;  period  as  a  base  at  which  all 
prices  are  reduced  to  1.00  or  100  and  with  which  all  sul)scqucut 
periods  are  compared  by  use  of  the  arithmetic  average  of  the 
price  deviations,  with  even  weighting,  have  also  worked  back- 
ward to  still  earlier  periods,  comparing  these  with  the  same  base, 
now  later  or  second  to  them,  still  using  the  arithmetic  average; 
and  have  then  strung  out  all  the  results  in  a  single  series.  We 
now  see  that  in  order  for  the  series  to  be  of  uniform  nature,  while 
the  arithmetic  average  is  used  in  com})aring  the  later  periods 
with  the  earlier  l)ase,  the  harmonic  average  should  be  used  in 
com|)aring  the  earlier  period.^  with  the  later  base  ;  or  conversely. 
Yet  as  these  statisticians  have  used  even  weighting  where  it 
does  not  belong,  it  does  not  much  matter  what  average  they  use. 

§  4.  Returning  to  our  argument,  we  see  that  if  the  continuous 
use  of  the  harmonic  average  over  two  periods  gives  a  result  too 
low,  each  single  use  of  it  gives  a  result  too  low  ;  wherefore  in 
the  first  examph',  when  the  Hrst  use  of  the  harmonic  average  in- 
dicated constancy,  there  was  really  a  rise  of  prices.  And  if  the 
continuous  use  of  the  arithmetic  average  over  two  periods  gives 
a  result  too  high,  each  single  use  of  it  gives  a  result  too  high  ; 
wherefore  in  that  example,  when  the  second  use  of  the  arith- 
metic average  indicated  cimstancy,  tiiere  was  really  a  fall  of 
prices.  And  reversely  in  the  se(!ond  example.  That  the  two 
kinds  of  averages  togetluir  give  tiic  right  result  at  the  end,  in- 
fallibly shows  that  each  one  alone  is  wrong — and  the  one  as 
much  wrong  on  the  one  side  as  the  otiicr  on  the  other.  Now,  to 
review,  in  our  first  example  the  indications  arc. 


TEST    CASES  o'29 

hy  tlie  Iiarinonic  average  :  hy  tlie  arithinetic  averatije  : 

for  the  1st  variation  1.00,  for  tlie  1st  vari;itioii  1.125, 
"       2d          "             .S8S8,  "       2d  "  1.00, 

"       whole   "  .8888;  "       whole   "  l.TJ-'i. 

Here  the  rio-ht  result  for  the  whole  comparison,  1.00,  is  the  (jeo- 
mctr'ie  mean  between  these  two  wrong  results.  Similarly  in  the 
seoimd  example  the  right  result,  1.00,  is  the  geometric  mean 
between  the  other  two,  1..33J  and  .To. 

Sup])ose  now  in  the  first  example  we  average  the  price  vari- 
ations geometrically.      The  geometric  mean  of  the  first  set  of 

.    .        .       \:\       3         '-*> 
price  variations  is  v      =  — '  =  l.OliOO,'   inJicatini''  a  rise 

of  6.06   per  cent.      The  geometric  mean  of  the  second   set  is 

2       4       1    8 

X   .,  =  ^^-  =  0.9428.-  indicating  a   fall  of  0.72  per  cent. 


J 


The  product  of  these  two  results  is  unity,  meaning  that  the  fall 
from  106.06  by  5.72  per  cent,  is  a  fall  to  1.00.  In  the  second 
example,  with  the  })rices  at  the  second  period  altered  to  the  arith- 
metic terms  around   1.00,  the  geometric  mean  of  the  first  set  of 

1    •> 
price  variatitms  indicates  a  fall  to      -  =  0.8()()0   by    13.40    ])er 

2 
cent.,  and   the  geometric   mean   of  the  second,  a  rise  to  = 

1    3 

1.1047  by  15.47  per  cent.     The  product  of  these  again  is  unity, 

the  rise  from  0.8660  by  15.47  per  cent,  being  a  rise  to  1.00.^ 

Thus  in  these  cases — and  the  universality  of  this  rule  is  proved 

in  Appendix  B  IV.  §  6 — the  confin.iioit^  itse  of  the  (/eoiiuiric  mean 

(/ire.s  f he  final  result  kiwirn  to  he  r'u/ht. 

Let  us,  again,  suppose  these  conditions  : 


I   100    A  @  1.00     100  B  0  1.00 
II     66tA@1.50     150  B@    .66| 
III  100    A  @  1.00     100  B  (7i)  1.00 


lOO  for  [A]  100  for  [B], 
100  for  [A]  100  for  [B], 
100  for  [A]     100  for  [B]. 


^  This  is  the  geonaetric  mean  between  1.00,  the  harmonic  mean  of  these  vari- 
ations, and  1.125,  the  arithmetic.  Therefore,  if  the  geometric  mean  of  the  price 
variations  he  the  right  one,  the  harmonic  mean  has  erred  as  much  helow  as  the 
arithmetics  has  erred  above,  these  proportions  being  measured  geometrically. 

-  This  is  the  geometric  mean  between  1.00,  the  arithmetic  mean  of  these  vari- 
ations, and  0.8888,  the  harmonic.     Therefore — as  in  the  preceding  note. 

■'Here,  too,  the  figures,  0.8660  and  1.1547,  are  geometric  means,  the  first  be- 
tween 1.00  and  0.75,  the  second  between  l.OO  and  l..!.'').'5o. 


:VM)  llli:    MHTIlol)    FOR    CONSTANT    SUMS 

The  ijeonu'tric  mean  nt"  each  (tf  these  sets  of  price  variations, 
.^,  1^,  and  |,  .^,  is  uiiitv  ;  and  the  I'oidt  toi'  the  whole  is  also 
constanev.  In  etieet.  imt  oidv  the  tii'st  j>riee  variations,  but  the 
second  likewise  are  uconietrie  variations  ;  for  at  the  second 
period  the  ])riee  of  %  \  is  l.OO  and  tlu'  price  of  H  B  is  1.00, 
and  at  the  third  period  the  prices  of  these  parts  of  A  and  B  are 
.()6|  and  l.oO.  Here  |  A  has  fallen  just  as  B  fell  in  the  first 
set  of  variations,  and  1  .^  B  has  risen  just  as  A  there  rose.  As 
regards  the  (piantities  purchased  with  every  two  money-units 
evenly  distributed,  at  the  second  period  compared  with  the  first, 
and  on  the  quantities  of  the  first,  they  are  |  A  and  |  B.  Thus 
in  the  first  quantity  variations  the  loss  is  of  J  A  and  the  gain 
of  J  B  ;  and  in  the  second,  reckoned  in  the  same  way,  the  gain 
is  of  I  A  and  the  loss  of  ^  B, — so  that  the  greater  gain  and  the 
smaller  loss  (as  a])pears  from  this  way  of  estimating  them)  fall 
alternately  on  the  one  and  on  the  other.  Therefore,  as  the  ul- 
timate result  of  the  two  geometric  averagings  is  correct,  we  have 
a  clear  advertisement  either  that  the  result  of  each  of  the  geo- 
metric averagings  is  correct,  or  that  the  error  in  each  is  corrected 
by  an  equal  error  in  the  other. 

But  in  the  preceding  examples,  in  which  we  dealt  with  only 
two  classes  equally  important,  the  geometric  mean  gave  a  result 
geometrically  midway  between  the  two  erroneous  results  given 
by  the  harmonic  and  arithmetic  averages,  whicli  averages  cor- 
rected each  other  at  the  end  of  the  second  variation,  showing 
that  they  diverged  e(iually  from  the  truth  on  each  occasion. 
Therefore,  in  these  cases,  undei-  these  conditions,  the  geometric 
mean  of  each  set  of  ])rice  variations,  coinciding  with  the  mean 
between  their  ecjual  errors,  was  exactly  correct.  And  therefore, 
in  general,  f/ie  (jcoineirk-  mean  of  price  variations,  whenever  it  is 
applicab/e,  i.s  correct. 

§  5.  So  far  we  have  dealt  with  only  two  ecpially  imj)()rtant 
classes.  Let  us  now  briefly  extend  the  investigation  to  cover 
uneven  weighting.     \\'e  may   suj)pose  the  followmg  conditions : 


1  100     A  Or,  1.00     -JOd     I",  f"    1.(1(1 
II    6(j|  A  O,)  I.TjO     2:y3l  n  Oi,    .8571  (: 
III  100    A  ((1)1.00     200    nOi)  l.OO 


1(10  for  [A]  200  for  [B], 
100  for  [A]  200  for  [B], 
10(1  for  [A]     '200  for  [B]. 


rr:sT  casks  'MM 

Here  tlie  liariiKniic  avcrauc  iiidieates  for  the  Hrst  priee  variations 
constaney,  for  the  second  a  fall  to  0.!>o3.')  bv  6|^  per  cent.  The 
aritlinietie  averag'e  iiidieates  for  the  first  |)rice  variations  a  rise 
to  1.0714  by  7.14  j)er  (;ent.,  for  tlie  second  constancy.  The  al- 
ternating use  of  these  two  averages,  in  either  order,  gives  the 
right  final  result.  Tlie  geometric  average  indicates  for  the  first 
price  variations  a  rise  to  i '  |^  =  l.O.'Vil)  by  o.29  per  cent.,  for 
the  second  a  fall  as  from  1.00  to  i''  ||  =  ().J>6<S1  by  ;3. 19  per 
cent.  The  continuous  use  of  the  geometric  average  gives  the 
right  final  result.  But  here  the  geometric  results  in  the  indi- 
vidual instances  no  longer  are  geometric  means  between  the 
other  two.  In  fiict,  \ve  now  have  no  reason  to  expect  even  the 
right  result  to  be  the  geometric  mean  betw'een  those  other  re- 
sults, since  we  are  now  dealing  with  higher  powers.  We  do, 
how^ever,  know  from  otn*  previous  reasoning  that  the  geometric 
average  is  not  exactly  right  in  its  single  uses.  Hence  the  other 
alternative  remains.  W  v  thus  have  demonstrative  confirmation 
of  our  previous  indu(;tions  ;  for,  as  the  final  result  is  right,  we 
see  (1)  that  the  error  in  the  f/eometric  averages  of  the  price  varia- 
tions must  he  above  the  truth  in  tlie  one  instance  and  beloiv  it  in 
the  other — and  the  determining  factors  can  only  be  found  in  the 
facts  that  in  the  one  instance  the  larger  class  has  fallen  in  price 
while  the  smaller  class  has  risen  and  in  the  other  the  larger  has 
risen  and  the  smaller  fallen,  although  the  connection  is  not  ap- 
parent ;  and  (2)  that  these  opposite  errors  must  he  (^geometricallt/) 
equal  to  each  other.  In  the  particular  case  before  us  these  con- 
clusions are  borne  out  by  applying  any  one  of  the  above-dis- 
covered foi-niuhe.      Any  one  of  these  gives  for   the  first   price 

.    .  .  307. <>U 

variations  a  rise  to  ^,._  ,.,.,.  =  1.0.'^3o  bv  3.35  per  cent.,  and  for 
'Ii)  t  .bhv  '  ^ 

•>97  0(59 

the  second    a   fall  as   from    1.00   to  .^V.^,  =  0.r>67()  bv  3.24 

30/.b34 

per  cent.     Thus  the  geometric  average  was  slightly  below  the 

truth  in  the  first  instance  (where  the  preponderating  class  fell  in 

price),  and  in  the  second  (where  the  preponderating  class  rose) 

slightly  above  the  truth  (so  that  we  may  induct  that  such  wdll 

be  the  connection  between  the  causes  and  effects  in   all  cases). 


'Sli'2  11  IK    MF/I'1I()1)     Foi;    ( ONSIA  NT    SIMS 

And  the  jreometric  indications  are  al)(>veaiid  below  the  truth  in 
e(|ual  i)r()i)ortions;for  l.O;*,:',.")  :  1.0;}2!)  ::  O.lMISl  :  ().9()7()  (with 
as  close  approximation  as  may  be  obtained  without  using  more 
decimals).     These  relations  will  be  found  to  hold  in  all  cases.* 

The  jjeneral  conclusion  from  this  investigation  is  that  in  deal- 
ing with  many  (ilasses  in  a  hmg  series  of  periods,  where  the 
many  tendencies  toward  neutralization  of  the  errors  have  room 
t^)  i)lay,  it  is  extremely  probal^le  tliat  the  result  given  by  the 
geometric  average  will  never  depart  nuieh  from  the  true  figure. 

But  of  course  this  conclusion  is  purely  theoretical,  since  the 
condition  presup])osed,  of  constant  sums  (or  sums  constantly  in 
the  same  proportion)  being  always  expended  on  all  the  classes, 
is  never  fulfilled  in  j)ractice. 

>^  <).  The  formula  above  dis(;overed  for  finding  the  general 
price  variation  between  fivo  periods,  in  all  cases  when  constant 
sums  are  expended  at  l)oth  j)eriods,  we  have  seen,  satisfies  this 
test  whenever  it  is  confined  to  a  revei'siou  of  the  state  of  things 
at  a  third  j^eriod  to  what  it  Avas  at  a  first  period  after  a  single 
deviation  at  a  second  period.  Now  suppose  that  the  same  state 
of  things  reverted  to  is  at  a  still  later  period,  after  three  or  more 
variations  through  two  or  more  intervening  periods.  Here  we 
have  a  test  for  this  formula  applied  in  a  series  of  measurements. 

Let  us  consider  the  next  simplest  case  ]>ossiblc,  illustrative  of 
these  complex  cases — consisting  of  two  classes  with  three  varia- 
tions extending  over  fi)ur  ])eriods.  We  may  j)osit  tliem  in  the 
universal,  or  algebraic  form,  as  follows : 

I  ^  A  0,}  «,      ^   1>,  (5^  /i,  I  a  for  [AT     b  f<.r  [B], 
1 1  ^   A  @  a.,      ^   B  @  ,:J,  I  a  for  [A]      b  Im-  [  F,], 

11 1  ^   A  @  a.,      ^   B  @  i\  ;  a  for  [  A  ]     b  for  [  B], 

"a  ;'3  i 

I  V   '^   A  @  «,       "   B  @  ;3,  1  a  for  [A]     b  for  [li]- 
"i  i\  I 

■'Till'  ar^mm'iit  liere  used  coiRU'riiiiig  tlu' geonu'tnc  iiicaii  ami  avfraj^i'  has  liccn 
suggested  i)y  Westergaard'.s  iirguineut  for  the  gcoiiictric  avoragi'  above  examined 
near  the  (muI  of  Sect.  VI.  in  Chap.  V.  lint  tiu;  tU'tails  and  tlie  conclusion  are 
dilVerent. — That  the  gooinetric  average  works  backwards  as  well  as  forwards,  has 
also  been  noticed  by  Wickscll,  I?.  l.S!',  p.  .S,  l)ut  without  pointing  out  th»»  condi- 
tions and  without  enijjloying  tiiis  fact  as  an  argument  in  favor  of  tlie  geinnetrio 
average  in  comparison  with  the  other  two. 


Ti:8T  CASES  .{;5;> 

As  the  conditions  aiv  exactly  the  same  at  the  f'onrth  jteriod  as 
at  the  first,  it  is  (ivideut  that  the  exchange- vahie  of  money  and 
the  level  of  prices  are  then  th{>  same  as  at  the  first  j^eriod,  and 
ought  to  be  so  indicated  in  the  series  of  measurements  in  w  Inch 
the  above  formula  is  applied  to  the  three  successive  price  vari- 
ations— as,  indeiKJ,  it  is  indicated  bv  a  direct  comparison  of  the 
fourth  with  the  first  period.  For  our  present  ])urj)ose  fi)rmula 
(6)  is  the  most  serviceable.  To  find  the  result  reached  through 
the  intermediate  com[xirisons  we  have  to  multiply  together  the 
three  applications  of  this  formula, — one  to  each  of  the  three  sets 
of  variations.  To  do  this  with  the  formula  in  its  full  form 
woidd  be  tedious.  We  may  simplify  the  operation  by  substi- 
tuting the  following  shortened  svmbols.    J^et  a  =     I— ,  b  =      |^', 

c  =  ^  \^,  (I  =  ^  '-^\  c  =  ^  I—",  f  =  ^  K^^       The    reciprocals    will 

\«;      \;V/      \,:^,  •     \l,t  ^ 

ecpial  the  reciprocals,  and  may  stand  for  them.  Also  he  =  <i, 
and  cj  =■  (h     The  formula  will  now  be 

/a     b\ 

p^      ^a/>  +  bo(ac  +  b/)(^-4-^^j 


P, 


a      b\  /a      b\  ,      ' 


which,  upon  multiplying  out,  reducing  and  rearranging,  be- 
comes 

(t  deb  f  (' 

„       a*-)-a'b  ,-hab-    -f-a-b  , -|-ab-    +a-b- -|-ab- ,. -)-b' 

^^=  ''  "  ''  ''  '*  -f  (<)) 

"p  1  ,  I  -f  '    v    ' 

'      a ' + ab-  ,  -I-  a-b  -  +  ab-  y  -^  a-b  -  -f-  ab^  -  -|-  a^b  -.  +  b^ 

<l  ((  h  ('  (■  f 

This  can  equal  unity  :  (1)  if  (t  =  <l,  b  =  c,  and  e  =  /",  that  is,  if 
there  are  no  price  variations  at  all,  or  if  all  the  price  variations 
at  any  or  every  stage  are  in  the  same  proportioli  ;  (2)  if  a  =  b, 
d=  e,  and  c  =/,  that  is,  if  there  are  no  price  variations  between 
the  second  and  third  periods  (which  virtually  reduces  the  series 
to  three  periods  with  two  sets  of  variations),  or  if  a  =  rb  and 
d  =  re,  while  c  = /,  that  is,  if  all  the  price  variations  are  alike 


:VM  iMi-;   Mirnioi)   koi;  (onstan'!'  scms 

between  the  seeoiul  and  third  periods  ;  (."))  it"a  =  b,  that  is,  it'  the 
two  classes  are  equally  iiu])(n'tant.  Phis  hist  eoiiditiun  cannot 
be  extended  t<^  cases  with  three  or  more  classes  indetinitely  ; 
but  it  mioht  be  extended  to  eases  with  4,  with  8,  with  KJ,  with 

:)2,  ehisses  ;  or  even   with   oth(>r   nuinl)ers  of  variously   im- 

])ortant  classes  if  tliesc  arranucd  themselves  suitably,  in  pairs  or 
sets.''  In  other  eases  the  fonnula  does  not  e(|ual  unity  unless  it 
hap]K'ns  to  contain  elements  that  eonuterbalauee  one  another  in 
tendinti'  to  deviations  in  opposite  direetious. 

Thus  in  this  simplest  exam|)le  of  the  complex  eases,  used  in  a 
series  of  only  four  ])eriods,  the  nu>thod  fails  to  stand  the  test 
universally,  even  if  every thiuii'  is  the  same  at  the  last  period  as 
at  the  first. 

§  7.  What  is  the  reason  for  the  faihu'c  of  this  method  in  a 
series  of  more  than  three  periods".' — fir  tiie  method  has,  appar- 
entlv,  been  demonstrated  to  be  true  in  every  sinij;;le  comparison 
of  two  contiguous  periods.  The  trouble  is  that  the  demonstra- 
tion has  l)een  directed  at  ])roving-  the  f  irnuda  to  be  true  in  every 
single  comparis(tn  of  two  eontiuuous  periods  only  in  reference  to 
each  other  and  out  of  couuectiou  with  any  other  |)eriods. 

Over  a  series  of  years  it  would  seem  as  if  the  perfectly  true 
method  should  be  modelled  on  what  we  have  done  in  the  case  of 
a  sintrle  measurement.  That  is,  we  should  seek  in  all  the  classes, 
for  service  as  imits,  masses  that  ai'c  e(pii\aleut  to  oiu'  another 
over  all  the  periods  compared,  and  should  nu'asure  at  every 
period  the  total  numbers  ol  these  mass-units  eoutained  in  all 
the  classes  at  that  period.  Ilepreseutiuii-  these  numbers  by 
x"',  ii'",  z'",  and  so  on,  it  is  plain  that  if  over  four  periods  we 
had  these  results  : 

•'■/"  +.'//"  +  -,'"  +  t"  "  terms  =  Q„ 

,r_^"  +  y/./"  +  -_/"  +  {n  n  terms  =  C^,, 

<"  +  .'/•/"  +  V"  +  to  /(  terms  =  %, 

■'•/"  +  //;"+V"  +  to  n  terms  =Q,, 

then  the  comparisons  (according  to  formula  1   in   Sect.  IIT.)  go- 

intj  through  this  series,  would  vield 

^  Cf.  the  coiiiUtions  for  the  coiTcrtiicss  of  tlic  ficonu'tric  average  above  pointed 
out  III  Sect.   I  I  I.  is.'). 


TKsr   (ASKS  .'Ulo 

1',     Q,    Q/ Q.,     Q,' 

just  as  if  wc  c'om})are(l  the  last  period  direc^tly  with  the  first. 
And  not  only  this,  but  also  the  comparison  through  the  serial 
forms  between  any  distant  periods  would  yield  the  same  re- 
sult as  would  a  direct  comparison  between  them,  no  matter 
what  be  the  states  of  things  at  these  periods  or  what  the  in- 
tervening variations,  jirovided  only  the  same  classes  be  used 
throughout.  Thus,  so  emended,  the  method  universally  satisfies 
Professor  Westergaard's  general  test.  It  is  not  difficult  to  show 
that  the  same  results  would  be  obtained  by  using,  between  any 
periods  in  an  epoch  of  n'  periods,  this  formula,  applied  to  any 
mass-units  and  their  prices, 


Pn'/  n'/ 

r.  1/    n  n  n   -X-  n  \/    i 


(10) 
x.y  a^a^a^ + //ai/'/^i^/^s +  

But  in  comparing  every  period  first  with  the  period  jireced- 
ing  and  then  with  the  period  fallowing,  if  the  state  of  things  at 
the  latter  period  be  irregularly  tliiferent  from  that  of  the  former, 
the  numbers  which  express  at  the  second  period  the  mass-units 
then  contained  that  are  equivalent  over  the  first  and  second 
periods  alone  are  not  the  same,  or  in  the  same  proportions,  as 
the  numbers  which  express  at  that  period  the  mass-units  then 
contained  that  are  equivalent  over  the  second  and  third  periods 
alone.  Therefore  a  direct  comparison  between  the  first  and 
third  periods  in  this  same  manner,  namely  by  com})aring  the 
numbers  contained  in  the  classes  at  each  period  of  the  masses 
that  are  ecpiivalent  over  the  first  and  third  periods  alone,  these 
numbers  being  again  different,  would  not  yield  the  same  result 
as  that  yielded  by  the  two  former  operations  together.  But  if 
all  prices  are  the  same  (or  in  the  same  proportion)  at  the  third 
as  at  the  first  period,  the  mass-units  used  in  both  the  compar- 
isons are  the  same,;  and  so,  in  such  cases,  there  is  necessary 

*•  Tlif  same  satisfactii)n  would  be  given  by 

P2  ^  «2^^xiA-2a;3  +/?2i/^yiy2yB  •••    + _ 

Pi      a^%/ x^x-iX.^ + /^i  K^  2/12/22/3 + ' 

but  this  woubl  not  agree  with  the  above  in  the  iiidividiial  results. 


336  TIN-:  Mi-yrnoD  for  constant  sims 

airreement  hetwet'ii  their  Hiinl  result  and  tlic  direet  eonipari.-oii 
between  the  first  and  thirtl  periods.  But,  again,  even  il'  all 
prices  are  the  same  at  a  fourth  or  hiter  j)eriod  as  at  the  first,  there 
has  been  a  break  in  the  (!ontinuity  of  the  mass-units  used  ;  and 
therefore  the  direct  comparison,  using  one  set  of  mass-units,  and 
the  indirect  comparison,,  using  two  or  more  other  sets  of  mass- 
units,  will  not  necessarily  coincide. 

The  direct  comi)arison,  however,  we  nmst  observe,  will  be  no 
more  authoritative  than  the  indirect  comparison  through  the  suc- 
cessive comparisons  of  the  intervening  periods.  The  operation 
which  uses  the  mass-units  that  are  equivalent  over  all  the  periods 
compared,  would  emend  both  those  oj)erations.  Yet  again,  that 
even  this  operation  would  be  authoritative,  does  not  appear.  For 
this  procedure  is  similar  to  that  of  using  the  geometric  average  of 
the  price  variations  with  the  same  weighting  over  all  the  periods, 
this  weighting  to  be  according  to  the  geometric  averages  of  the 
money-values  at  all  the  periods — a  j)rocedure  against  which  ob- 
jection has  already  been  raised." 

Moreover,  in  practice  it  is  iuipossil)le  to  emj)loy  this  con- 
sistent method  in  a  long  series,  especially  as  it  would  have  to  be 
revised  in  full  every  year  upon  the  adding  of  a  new  period.  We 
are  practically  confined  to  the  course  of  com])aring  every  period 
with  the  |)eriods  immediately  preceding  and  following,  and  of 
comparing  distant  pei'iods  through  the  mediaticm  of  such  inter- 
vening measurements.  It  is  desirable,  then,  to  form  an  idea  of 
the  possible  extent  of  the  eri'or  in  this  procedure,  and,  if  pos- 
sible, of  tUv  pi'ineiples  that  gctvern  it. 

§  8.  Suppose  the  class  [B]  is  twice  as  large  as  the  class  [A], 
or,  in  other  terms,  a  =  1  and  b  =  2.      Then  formula  (9)  becomes 

Again,  suppose  [B]  is  three  times  as  largcas  [A].     Then 

■  111   (  'liiipt.   \'.  i  y\ .  IH'Mr  rlid. 


TKST    CASKS 


:VA7 


The  divergent  ])arts  of  the  expressions  are  in  the  hist  three  terms 

in   the  nnmerators  and   denominators.      If   [B]  be   four   times 

larger  than   [A],  tlie  eonnnon   terms  in   the  Ijraekets  wonkl  be 

(I  <1 

1  +  4   ,  4-  -i     +  +  'i^,  and  the  three  terms  in  excess  wonld 

((  a 

(J       h        r  \ 
"        'J  J 


each  have  12  for  their  coefficient.      Let 


resented  })y  E',  and  I    /+/  +  /)  ^^X  1'^"-    Then,  with  b  =2  a, 
the  first  of  the  above  fornndfe  may  be  expressed  in  this  simple 


manner, 


P,       [41 +  E'  +  E"]+E' 


■  Pj       [41  +  E'  +  E"]  +  E" ' 

and  with  b  =  '■]  a,  the  other  may  be  expressed  thus, 

P,  _  [9J^-f  E'  +  E'']  +  2  E' 
Pi  ~  \y>J  +  E'  +  E"]  ^2W  ' 

and  with  b  =  4  a,  the  formuki  would  be 

P,  _  [161 +  E^  _^E^^]  +  3  E' 
P;  ~  [l(j  1  +  E' +  E^]  +3E"  ' 

and  so  on.      In  general,  therefore,  letting  a  =  1  and  b  =  r,  the 
comprehensive  formula  will  be 


p  +  ^  +  E'  +  E'' 


['•.; 


+  E'  +  E" 


+  (r-l)E' 


+  (7--l)E" 


01) 


or 


r2  +      ^  rE'  +  E" 


1^     ,.2+  _  +E'  4-^-E" 


(12) 


On  the  other  hand,  if  [A]  be  the  larger  class,  everything  will 
work  out  merely  to  the  inverse  of  the  preceding,  so  that,  with 
a  =  /•'  and  b  =  1 ,  the  universal  formula  is 


.■J.j8  iiii;  .Mi:iii()i>   I'oi;  (onstant  sims 

P 

In    these  general    ionnula',  if    /•  (oi-  /•' )  =  1,  then  p' =  1.00; 

1 
wliieh    i.s   in   aiircenient  with    the   third  ('(indititui  al)i)ve   noticed 

for  tlie  rednetion  ot"  fornuda  (10)  to  ninty.      Auain,  as  /•  or  r' , 

I' 
increases   toward    infinitv.   the    i-c-nlt,    yy' ,    a|)jiroaehes    toward 

I 

unity.      This   is   in   aiiivcnicnt   with  what   was  to  be  expected; 

for  as   the  one  elass  infinitely  j)redonnnates,  the  other  sinks  to 

nothingness,  and  tlie  nieasnrenient  is  virtnally  by  one  class,  but 

indirect  (serial)  com])arisons   with   one  class  will   always  agree 

with  direct  comparisons  with  that  class.      These  two  facts  show 

that  somewhere  between  equality  in  size  of  the  two  classes  and 

the   infinite  predominance  of  the  one,  tliere  is  an   ine(|uality  at 

which  the  error  is  at  its  maxiinum.      Now   from  the  wav  FJ  and 

E"  are  constructed,  it  is   plain  that  they  nuist  always  be  ahonf 

'■'>.      This    being  so,  it  is  easy  to  find   that  the  greatest   possible 

error  must  always  happen  when  the  one  elass  is  ;ibout  ton  r  times 

as  large  as  the  other.      Ivoughly,  also,  it  may  be  perceived,  the 

greatest  ])ossibl(M'rror  will  lie  between  one  third  and  one  fourth 

K'  /       E"  \ 

of  the  diirci-cncc  between  ^^,,  I  oi'  .,,   .  as  the  ease  ina\"  be  j  and 

unity. 

Consider  the  following  example  : 

I  lUO     AOol.(H)  200    JU"   1.0(1  loo  t(,r  [A]  200  lor  [  F.], 

II  6Q'i  A  @  1.50  2()()J!  B(^    .7.")  100  for  [A]  200  for  LB], 

III  50     \07:2.O()  .".00    V,(„     .(iC.H  100  for  [A]  200  for  [B], 

I\'  100     A  r-'    1.00  200    I',  r-    1.00  100  for  [.\]  200  for  [R]. 

1* 

To  find      '  as  reached  through  the    interxcning   comparisons,  we 

need  only  calculate  out   !'>'  and   K"   and  a|)|)l\    them    in    fornuda 

(ll)()r(r2).      They  are    found    to    be   ."'..•J  1  (i:'>    and    .■')._!.")")(;    re- 

I*        1  i.iss-j 
.s|)e(;tivelv  ;   whence        =  =  (».!»!l72,  indicatinti' a  fall  of 

I  ,         1  4 . 1:  ■_!  /  • ) 


TEST    CASES  ."5  •■*)'.< 

O.'iS  per  cent.,  and  committini;-  an  en-or  to  that  extent.  If, 
keeping-  the  same  price  variations,  we  sn})p()se(l  the  mass- 
quantities  to  be  such  as  to  make  the  class  [B]  constantly  three 

P,       22.-i37<S 
times    larger    than     [A],    we    should     have    p  =  ,j_^  .  = 

P, 

(».!)!)()47  ;  if  [B]  were  four  times  larger  than   [A],  then  .,    = 

I 

^-^^^—=0.99687;   if   [B]    were  five  times  larger  than   [A], 

P       44  5371 
then  ^'  =  ttWto  =  0.99648  :  —  and  thereafter  the  result  will 
Pj       44.D94f3 

rise  toward  unity.     Thus  in  this  case  the  greatest  possible  error 

E' 

is  al)out  0.3(>  per  cent,  below  the  truth.      Here  ^„  =  0.9879, 

which  is  1.21  percent,  below  unity,  and  about  3^  times  as  large 
as  the  greatest  possible  error. 

If  these  examples  were  turned  about,  the  same  price  varia- 
tions being  kept,  but  the  class  [A]  being  supposed  the  con- 
stantly larger,  the  results  would  lie  the  reciprocals  of  the   ])re- 

P  V 

ceding.      Thus  with  a  =  2  b,  -"  =  1.0028  ;  with  a  =  3  b,  p'  = 

*  1  1 

P  P 

1 .00353  ;  with  a  =  4  b,  ^^  =  1 .00363  ;    with   a  =  5  b,    p'  = 

1.00352.  The  maximum  error  is  about  0.36  per  cent,  above 
the  truth. 

In  the  former  of  these  examples  the  erroneous  results  were 
below  the  truth,  following  the  gradual  fall  from  the  first  period 
of  the  larger  class  ;  and  in  the  latter  they  were  above  the  truth, 
following  the  gradual  rise  from  the  first  ])eriod  also  of  the  larger 
class.  If  we  inverted  th(!  intermediate  periods,  so  that  the 
third  came  second  and  the  second  third,  the  values  of  E'  and  E" 
would  change  places.  Then  in  the  fi)rmer  exam])]es  (with  [B] 
larger)  the  erroneous  results  would  be  above  the  truth,  follow- 
ing the  gradual  rise  from  the  second  period  of  the  larger  class  ; 
and  in  the  latter  examples  (with  [A]  larger)  the  erroneous  re- 
sults would  be  below  the  truth,  following  the  gradual  fall  from 
the  second  period  still  of  the  larger  class. 


;i4() 


TJII-:   Mi:rii()i)   I'oi;  coNsTANr  sims 


If  we  make  eaeli  class  hotli  rise  and  fall  in  price,  we  u'<'t  the 
most  satisfactorv  i-esiilts  hy  niakinii'  tlieni  rise  and  fdl  toucrlier, 
as  ill  the  followinu- : 

I   100    A  @  l.OU  200    B  @  1.00  j  100  for  [A]  'iOO  for  [B]. 

ri     ()()iA@1.50  100    B  @  2.00    100  for  [A]  200  for  [B], 

III  150    A@    M'n-  2661  B  @    .To    100  for  [A]  200  for  [B]. 

IV  100    A  (^  1.00  200    B  @  1.00  |  100  for  [A]  200  for  [B]. 

P 
Here  E'=  iLOiry,]  and  E"=  :].01G1,  whence      '=  O.iUtiMM,  in- 

I 
dicating-  a   fall   of   O.OOl)    per  cent.,  with   a   j)(»ssil)ilit\"  of  error 

E' 

throiio'h  larger  size  of  [B]  shown  by  ^p^,  =  0.1)nJ»7  not  to  ex- 
ceed 0.01  per  cent,  below  the  truth.  The  class  [A]  beinjj;  the 
larg-er,  the  results  would  be  inverted, — as  also  were  the  interven- 
ing })eriods  alternated.  And  we  get  the  least  satisfactory  results 
by  making  the  classes  rise  and  fall  in  price  oppositely, — thus  : 


I  100    A  @  1.00  200    B  @  1.00 

II  66s  A  @  1.50  2661  B  @    .75 

III  150    A  @    .661  100    B  @  2.00 

TV  100    A  @  1.00  200    B@1.00 


100  for  [A]  200  for  [B]. 

100  for  [A]  200  for  [B], 

100  for  [A]  200  for  [B], 

100  for  [A]  200  for  [B]; 

I^ 


for  here  E'=  3.5545  and   E"=  3.7389,  whence  ^'  =  0.98S4, 

wrongly  indicating  a  fall  of  1.1(>   per  cent.      With    [B]  three 

P 
times  as  large  as  [A],  cj  =  0.9851  ;    with   [B]  four  times  as 

P 

large    as  [A],  p-*  =  <).9S45;    with    [!>]   five  times    as    large    as 

1*  P 

[A],    -~  =  ().9S4()  ;    and    thereafter     p'  grows   greater   toward 

'  1  1 

unity.      Thus  the  error  may,  in  this  extravagant  examj)le,  be  as 

great  as  about  1  .'^  percent.     Again,  if  [A]  w<'re  the  larger  class, 

the  residts  would   merely  be  inverted, — as  als(»   would   be  the 

case,  again,  if  the  second  and  third  periods  were  reversed. 

In  a  series  of  live  periods,  the  fornnda^  would  be  still  longer; 
and  longer  still  in  a  series  of  six  periods,  and  so  on.  These  be- 
come too  complex  to  M'ork  out.  \\ C  may  I'cly  on  trial.  And 
trial  still  shows  but  slight  errors. 

In  the  above  examples  it  is  difficult  to  see  a  definite  principle 


TKST    CASKS  .MI 

(Ictefiiiiiiiii^  w  liicli  way  tlic  error  sliall  uo,  or  liow  far.  A  state- 
ment more  than  oiiee  made  that  ///  a  scries  of  four  jx'fiods^  If  the 
.second  (did  third  cIkiih/c  p/ticcs^  fhr  finaf  result  is  inverted,  inav  ho 
seen  to  he  uni\-ersal,  hy  ins|)eetiii<>-  the  universal  f'orniulie.  In 
a  lonn'cr  series  tliis  i-ide  can  :i|)|)ly  only  to  the  alternation  of 
periods  e(|nally  distant  from  the  first  and  the  last.  Tt  is  certain, 
then,  that  theiv  is  nothiny;  determining;  a  continual  error  in  one 
direction.  And  the  errors,  so  small  in  the  above  extraordinary 
cases,  will  |)rol)ahiy  he  hut  triflin<>-  in  ordinary  cases.  In  a 
general  way  there  is  some  resemhlance  to  the  erroncousness  al- 
ready found  in  the  geometric  averaging  of  the  price  variations, 
although  the  errors  are  prohal)ly  smaller  here.  When  manv 
classes  are  dealt  with,  there  is  no  reason  to  suppose  the  errors 
will  accumulate,  hut  rather  is  there  jirobahility  they  will  tend  to 
neutralize  one  another.  Also  if  the  result  hap])ens  to  be  above 
the  truth  at  the  fourth  |)eriod,  it  is  as  likely  as  not  to  be  below 
tiie  truth  at  the  fifth  period,  there  being  balancing  from  period 
to  period.  There  is,  then,  extreme  ])robability  that  in  a  long 
series  this  method  will  correct  itselfj  and  the  results  indicated 
by  it  will  always  hover  in  the  neighborhood  of  the  truth. 

But  the  fact  that  this  method  shows  error  at  all  in  a  series 
of  four  or  more  j)erio(ls — not  to  mention  inconsistency  in  a  series 
of  three  ])eriods — casts  susj)icion  upon  its  perfect  correctness 
even  in  a  comparison  of  two  periods.  And  yet  the  reason- 
ing by  which  it  has  been  discovered  has  seemed  to  be  sound. 
The  method  will  later  be  involved  in  a  complete  method  for  all 
possible  cases.  Then  will  be  the  occasion  to  continue  investi- 
gating the  fault  in  it. 


CHAPTER   XI. 

THE  Mpynioi)  FOK  CONSTANT  MASS-QUANTITIES. 
I. 

§1.  As  in  the  case  of  tlie  preceding-  argument,  so  the  argu- 
ment from  compensation  by  equal  sums  of  mon(>y  has  been  seen 
to  be  applicable  to  all  three  averages  not  only  when  the  subject 
of  weighting  is  left  out  of  account,  but  also  when  this  is  taken 
into  consideration.  The  appearance  of  application  solely  to  the 
arithmetic  average  is  due  to  the  selection  of  the  mass-units  in 
which  the  prices  are  reckoned  ;  for  attention  is  often  transferred 
from  compensation  by  equal  sums  to  compensation  by  equal 
prices. 

In  a  certain  way  by  compensation  by  equal  sums  spent  on 
constant  mass-quantities '  is  meant  a  compensation  by  equal 
price  variations  ;  but  it  is  an  equal  variation  in  the  price  pri- 
marily of  the  it^lioJe  (juantity  of  every  class  that  is  purchased  at 
each  period.  Tiius  if  we  represent  the  prices  at  the  two  periods 
of  the  constant  whole  quantity  of  [A]  by  a,  and  a., ,  and  those 
of  the  constant  whole  quantity  of  []3]  l)y  b^  and  b., ,  and  con- 
fine our  attention  to  these  two  classes,  the  position   underlying 

p^     a.,  -h  b., 
the  present  argument  postulates  that  j^'  =    ~         ~ ;  and  for  con- 

1 ,      aj  4-  D, 

stancy  of  the  j)rice-level,  requiring  that  a.,  +  b„  =  a,  -|-  b, ,  it 
requires  (supposing  A  rises  and  B  falls  in  price)  that  a.,  —  a, 
=  b,  -  b, . 

But  the  whole  masses  of  [A]  and  [B]  may  be  divided  into 

'  As  ill  the  preceding  Chapter,  all  that  is  said  here  is  equally  applicable  to  cases 
in  which  at  hoth  periods  are  purchased  mass-quantities  in  the  same  proportion, 
whether  at  tiic  second  period  they  ix^all  larger  or  all  smaller  than  at  the  first,  pro- 
vided all  the  niasoning  (and  all  the  fornuilaj  later  to  be  described)  be  applied  only 
to  the  mass-quantities  of  the  one  or  the  other  period — or  only  to  what  is  common 
to  both  periods.     t,'f.  also  Appendix  A,  YII.  ^5. 

.342 


( o.MrKNsA  ii(».\    \;\    i:<^)rAL  sr.\is  AAll 

any  miinhers  of  inass-iiiiit-  ot"  various  sizes,  the  priees  of  \vliu;h 
are  the  ones  eited.  ircncc  for  c  nistaney  of  the  priee  level  it  is 
only  the  j)i'iee  variations  of  tlic  a<^<i;rei»;ates  of  the  mass-units 
that  arc  i-c(|uir('il  to  ('(lual  each  other, — not  the  |»artienlar  ])riee 
variations  that  happen  to  be  reported.  Tiiis  is  evident  when 
we  reflect  that  the  one  class  may  he  mnch  larger  and  more  im- 
portant tlian  the  other,  in  which  case  the  priee  variation  of  any 
individual  in  the  former  is  re(|uired  by  every  onv  of  the  aver- 
ages to  be  smaller  than  tiiat  of  an  ex^mxl  individual  in  the  latter. 
For  the  present,  however,  we  shall  supi)()se  ourselves  to  be  deal- 
ing- with  ecjnally  large  or  im|)ortant  classes,  so  as  to  have  the 
convenience  at  first  of  dealing  with  even  weighting. 

Now  when  the  whole  mass-quantities  of  every  class  purchased 
at  each  period  are  constant,  but  there  are  variations  of  prices,  we 
have  seen  that  the  weights  of  the  classes  are  different  at  the  two 
periods,  because  the  weights  are  according  to  the  total  exchange- 
values  (or  money-vahies)  of  the  classes,  and  these  must  have 
varied  under  the  conditions  sup|)oscd.  Jlence,  wanting  to  use 
ev^en  weighting,  we  liave  tiic  o[)tion  l)etween  three  systems:  (1) 
even  weighting  of  the  fir-^f  period,  (2)  even  weighting  of  the 
second  period,  (3)  even  weighting  of  hof/i  the  periods  together. 
If  it  happens  that  the  total  money-values  of  the  classes  are 
equal  at  the  first  period,  then  the  equal  compensation  in  their 
variations  are//'o/»  e([uality  to  equally  distant  opposite  positions 
— which  we  know  to  l)e  arithmetic  variations.  If  it  happens 
that  the  total  money-values  of  the  classes  are  equal  at  the  sec- 
ond period,  then  the  equal  compensation  in  their  variations  are 
from  equally  distant  opposite  positions  to  equality — which  we 
know  to  be  harmonic  variations.  If  it  happens  that  the  total 
money-values  of  two  classes  are  alternately  equal,  that  of  the 
one  at  the  first  period  being  equal  to  that  of  the  other  at  the 
second,  and  reversely,  then  the  equal  compensation  in  their 
variations  are  from  opposite  positions  to  reversely  opposite  posi- 
tions, traversing  the  same  road  in  opj)osite  directions, — which  we 
know  to  be  ge(jmetric  variations. 

If,  further,  we  divide  the  total  masses  of  each  class  into  ideal 
mass-units  that  are  equivalent  at  the  first  period,  then  if  the 


344  rill-:   .MirriioD   i-oi;  coNs'iAN'r  massks 

total  inoncy-valuos  of  the  classes  happen  to  be  e([ual  at  the  first 
period,  and  the  variations  of  these  are  to  the  opposite  arithmetic 
extrenies,  the  variations  of  the  prices  of  these  mass-units  will 
also  l)e  to  the  oj>posite  arithmetic  extremes.  But  if  the  total 
money-values  happen  to  be  e({ual  at  the  second  period  and  their 
variations  arc  from  the  op])osite  arithmetic  terms,  although  we 
have  compensation  by  ecpial  sums,  we  do  not  in  this  case  have 
compensation  by  equal  prices  of  these  mass-units,  Avhich  have 
\aried  to  the  o})posite  harmonic  extremes.  And  similarly  there 
is  no  equal  compensation  in  the  prices  of  these  mass-units  when 
there  may  be  such  compensation  in  the  sums,  if  the  total  money- 
values  happen  to  be  equal  over  both  the  periods  together.  Hence 
in  these  cases,  although  the  compensation  by  equal  sums  really 
exists  as  well  as  in  the  first  case,  yet  the  com])ensation  by  equal 
variations  of  the  prices  that  happen  to  be  cited  has  disappeared. 
This  is  why  the  argument  under  consideration  has  seemed  to  be 
an  argument  ])eculiarly  in  favor  of  the  arithmetic  averaging  of 
])rice  variations ;  for  statisticians,  as  we  know,  have  formed  the 
habit  of  using  the  variations  of  prices  equal  at  the  first  period. 
But  if  we  used  mass-units  equivalent  at  the  second  ])eriod,  or 
over  both  the  periods  together,  we  should  get  compensation  by 
equal  prices  (of  such  mass-units)  even  when  the  price  variations 
are  the  harmonic  or  the  geometric.  AVe  need  now  to  j'cview 
these  possible  positions. 

^  2.  Using  mass-units  that  are  equivalent  at  the  first  period, 
we  may  construct  tlie  following  schemata  illustrative  of  the  con- 
ditions when  there  is  compensation  by  ecpiality  in  the  variations 
of  the  total  money-values  of  the  total  mass-quantities — in  the 
cases,  to  begin  with,  of  two  classes  supposed  to  be  equally  im- 
])ortant  at  some  pei-iod  or  periods.  The  only  changes  in  the 
construction  of  these  schemata  from  those  in  the  last  (chapter  is 
the  omission  of  the  sums  of  the  mass-units  at  each  ])ei"iod,  and 
instead  the  insertion,  at  the  extreme  right,  of  the  t<»tal  sinus  of 
money  expended  at  each  period  on  both  the  classes.  Thus  on 
mass-units  ecpiivalent  at  the  first  period  we  have  compensation 
by  equal  sums  when  the  price  variations  are  to  the  simple  arith- 
metic extremes,  as  follows  : 


(•0>(l'F,NSA'ri<)N    \',\    i;(;iAI,    SIMS  ."Ho 

I  10(1     \  ('<    1.<K(      100    \\(„    1.00  1   100  for  [A]      100  lor  [R]    -200, 
II  100    A  C"    l.oO     100   IW"     ..")0  I   l.'O  for  [A]        ■')()  for  [P.]  —  200; 

when  the  price  variations  are  to  the  siniph'  harmonic  extremes: 

I  100  A  (at  1.00     200  I!  (o    1.00  I   100  for  [A]     2(t0  for  [15]  —800, 

II  100  A  Qv.  1.50     200  r.  (a     .7")  |   l-'iO  for  [A]      L'jO  for  [R]  — :U)0; 

when  the  price  variations  are  to  the  sinij)h'  oconietrie  extremes  : 

I    100  A  (»•:.  1.00     L'lO  1!  (-/    1.00     I  100  for  [A]      l')0  for  [B]  —  250, 
II   100  A  Or  !..-)()     ].-)0  P,  0,     .(10:^1  I  l.-)0  for  [A]      100  for  [B]— 250." 

Here  it  is  onh'  in  tiietirst  selienia  that  we  have  compensation 
by  e(jual  prices;  but  in  all  of  them  we  have  compensation  by 
equal  sums.  We  find  also  here  the;  peculiarities  with  which  we 
are  familiar.  In  the  first  the  sums  are  equal  at  the  first  period, 
and  the  compensation  hy  e((nal  sums  is  away  from  this  ecpial 
condition.  In  the  second  the  sums  are  ecpial  at  the  second 
period,  and  the  compensation  is  by  equal  variations  froiu  the 
first  condition  to  this  e<[ual  second  condition.  In  the  third  the 
equal  sums  alternate  and  chan<i'e  places,  traversing-  not  only 
equal  distances,  but  the  same  road. 

That  these  peculiarities  are  universal  may  likewise  be  pi'oved 
l)v  the  formula  discovered  in  §  o  of  Section  II.  in  (Jhapter  IX. 
In  that  formula  the  mass-units  were  supposed  to  be  e(piivalent  at 
the  first  period,  wherefore  it  is  ajiplicable  here.  T>etting  the 
price  of  A  always  be  supposed  to  rise  from  1.00  to  a./,  we  know 
that  if  the  price  of  !>  falls  fi-om  1.00  to   the  opposite  arithmetic 

extreme  it  falls  to  2  —  '/,',  if   to  the   iiarmonic,  to    ,     ."      -,  '   if 

- '/.,   —  1 

to  the  i^eometric,  to      ,  •      Snp[)lying  these   values  of  ji.^'   in  the 

formula 

a,'  -  1 

*  We  might  also  here  lUiike  the  total  siiiiis  always  add  up  to  the  same  figures, 
as  follows  for  the  second  and  tliird  : 

I6t)!5A@l.O0     l.-iHi  Bf't  1.00  I    (JliH  for  [A]     133^  for  [B]— 200, 
11  Ufii;  A(g|l..->0     U-XhBOv    .7.')  |  100    for  [A]     100    for  [B]— 200; 

1  80  A®  1.00     120  I?  (m  1.(1(1    I    so  for  [A]     120  for  [H]— 2(»0, 
II  SO  A@  !..')()     120  B  (nj    .(Kin  |  120  for  [A]       SO  for  [B]— 200. 

But  here  again  the  differences  in  the  sums  arc  ditfercnt.     Cf.  Note  2  in  Sect.  I.  of 
the  last  Chapter. 


34()  'PI IK  >Fr:TFi()D  for  coxstant  masses 

wf  iind.  when  the  |)iM('c  viirintioiis  are  to  the  afithiiictic  ex- 
tremes, 

wlien  tliey  are  to  tlic  liarnionic  extremes, 

when  they  are  to  the  ii'eometric  extremes, 

The  first  of  these  expressions  means  that  when  tlie  priee  vari- 
ations are  to  the  opposite  arithmetic  extremes,  in  order  to  have 
compensation  by  equal  sums,  we  must  purchase  an  equal  number 
of  mass-units  of  [A]  and  [B]  ;  but  these  mass-units  are,  by 
hypothesis,  e(piivalent  at  the  first  ^^eriod  ;  therefore  we  must 
spend  equal  sums  on  [A]  and  [B],  or  spend  our  total  snm 
evenly,  at  the  first  period.  This,  moreover,  is  ev'ident,  becuuise 
the  prices  of  these  mass-units  being  supposed  to  vary  t«  the  op- 
posite arithmetic  extremes,  it  is  only  by  these  mass-units  being 
purchased  in  equal  numbers  that  the  total  sums  will  vary  at  the 
second  period  to  the  opj)osite  arithmetic  extremes.  The  second 
of  these  expressions  means  that  when  the  price  variations  are  to 
the  opposite;  harmonic  extremes,  in  order  to  have  compensation 
by  equal  sums,  we  must  purchase  for  every  mass-unit  of  [A] 
2  a.'  —  1  mass-units  of  [B]  ecpiivalent  at  the  first  period,  that 
is,  for  every  money-unit  spent  on  [A]  at  the  first  period  we 
must  spend  this  number  of  money-units  on  [B]  at  that  ))criod  ; 
but  at  the  second  period  we  must   spend  a..'  on   this  1  A  and  on 

these  (2  a!  —  1)  B's  we  nnist  spend  (2  a..!  —  I)-,      :       ,  =  «-', 

that  is,  we  must  spend  ecpial  sums  on  [A]  and  [B],  or  sj)end 
our  total  sum  evenly,  at  the  second  j)eriod.  Tlie  tliird  means 
that  when  the  price  variations  are  to  the  o|)])osite  geometric  ex- 
tremes, in  order  to  have  compensation  by  e(pial  sums,  we  must 
purchase  at  the  first  period  for  every  1  A  «./  B's,  and  these  mass- 
units  being  equivalent  at  the  first  period,  we  must  spend  our 
money  in  these  proportions  at  the  first  period,  that  is,  a^'  nioney- 
units  on    [B]    for  every  one  money-unit  on  [A]  ;  but  at  the 


CO.Ml'KNSATION     I'.V     i:(irAI,    SUMS  ."HI 

second  |)('ri(t(l  to  imrcliasc  the  1   X  we  have  to  speiKl  a,'  luoney- 

uuits,  and  to  i^ircha'^e  the  a,'  I>'s  we  have  to  spend  a.,'  ■       ^  =  1 

inoney-unit,  tliat  is,  the  sums  must  at  tlie  second  period  be  th(; 
reverse  of  \vhat  they  were  at  tlie  first.* 

§  3.  If  we  use  different  mass-units,  the  formulae  and  the  re- 
sults for  I/"  (the  ratio  between  the  mass-units  in  the  classes) 
will  be  different ;  but  the  results  above  obtained  for  the  total 
sums  to  be  spent  on  [A]  and  [B]  at  each  period  will  not  be  af- 
fected.    Therefore  these  results  are  universal. 

We  may,  tlien,  pass  quickly  over  the  schemata  for  the  condi- 
tions when  we  use  mass-units  equivalent  at  the  second  and  over 
both  periods.  These  may  be  easily  obtained  from  the  preced- 
ing by  rearrangement  of  the  mass-units,  and  their  prices,  of 
[B] .  Thus  with  mass-units  equivalent  at  the  second  period,  we 
have,  for  the  arithmetic  price  variations  : 

IU)0A@;1.00       33iB^@3.00     100  for  [A]     100  for  [B]— 200, 
lI100A@l.o0       33iB'@1.50     150  for  [A]       50  for  [B]— 200; 

for  the  harmonic  price  variations  : 

I  100  A  @  1.00    100  B'  @  2.00 
II  100  A  @  1.50     100  B^  @  1.50 

for  the  geometric  price  variations 

I  100  A  @  1.00     66t  B^  @,  2.25 

II  100  A  @  1.50    66|  B'  (a)  1.50 

Here  the  particular  and  total  sums  are  the  same  as  in  the  pre- 
ceding examples,  only  the  numbers  of  the  mass-units  of  [B] 
being  changed.  It  is  now  only  in  the  harmonic  price  variations 
that  compensation  takes  place  by  equal  prices. 

Arith  mass-units  equivalent  over  both  periods  we  have  for  the 
arithmetic  price  variations  : 

I  100  A  (w;  1.00     57.73  W  @  1.7320  I  100  for  [A]     100  for  [B]— 200, 
II  100  A  @  1.50     57.73  B"  @    .8660  |  150  for  [A]       50  for  [BJ— 200; 

'  111  this  ease,  if  out  of  2  lUDiicy-units  we  spend  a^    times  more  on  [B]  .than  ou 

[A]  at  the  first  period,  we  must  tluii  spend  r ZT^  **"  t-^^  '"**'  '  -  i  ""  t''^- 
and  at  the  second  ])eriod  the  reverse.     Tiiese  figures,  as  before,  at  arithmetit; 

extremes  around   1,  are  the  harmonie  me:uis  between  1  and  —7  (1.  e.,  do')  and  1 

"2 

and  a./  respeetively.     Cf.  Note  .'5  in  Section  I.  of  the  hist  chapter. 


100  for  [A]     200  for  [B]— 300, 
150  for  [A]     ^^^  for  [B]— 300; 


100  for  [A]     150  for  [I?]— 250, 
150  for  [A]     100  for  [B]— 250, 


100  for  [A]      150  for  [P,]— "ioO, 
]r>()  for  [A]     100  for  [B]— 2o0. 


348  TIIH    MF/niOl)    l-oi;    COXSTANT    MASSES 

for  the  liannonie  j)rice  variations  : 

I  100  A  @  1.00     111.42  B'''@  1.4142  I  100  for  [A]     200  for  [B]— 300, 
II    100  .V  Oi},  1.50     141 .4-2  B"  Oi    l.OfiOC.  |  1.50  for  [A]      150  f„r  [B]-  800; 

for  the  i^eometric  price  variations  : 

I  100  A  @  1.00    100  B''  (a)j  1.50 
II  100  A  @  1.50     100  Y,"  (5'  1.00 

And  here,  too,  the  particnlar  and  total  sums  are  the  same  as  be- 
fore, only  the  numbers  of  the  mass-units,  and  their  prices,  of 
[B]  being  changed.  And  it  is  now  only  in  the  geometric  price 
variations  that  compensation  takes  place  by  equal  prices, 

II. 

§  1 .  Now  the  argument  from  com])ensation  by  equal  sums 
must  claim  that  in  every  one  of  the  three  price  variations  above 
thrice  schematized  the  exchange-value  of  money,  or  the  general 
level  of  prices,  remains  constant,  because  this  compensation  ex- 
ists in  them  all.  But,  in  each  of  the  three  sets,  in  the  first  ex- 
amples, which  always  illustrate  the  arithmetic  price  variations, 
the  arithmetic  average  indicates  constancy  only  with  even  weight- 
ing— and  this  is  the  weighting  only  of  ihe  first  period.  In  the 
second  exam])les,  always  illustrative  of  the  harmonic  price  varia- 
tions, the  harmonic  average  indicates  constancy  only  with  even 
weighting — and  this  is  the  weighting  only  of  the  second  period. 
In  the  third  exam])les,  illustrative  of  the  geometric  jirice  varia- 
tions, the  geometric  average  indicates  constancy  oidy  with  even 
weighting — and  this  is  the  weighting  only  of  both  the  periods 
together.  And  these  conditions  we  know,  by  means  of  the 
formula  above  enq)loye(l,  to  be  universal. 

We  find  also  that  if  we  use  the  arithmetic  average  upon  every 
single  one  of  these  schemata  with  the  weighting  of  the  first 
period,  which  is  always  inieven  in  the  second  and  third  ex- 
amples in  each  set,  we  always  get  indication  of  constancy ;  that 
if  wc!  use  the  harmonic  average  upon  every  single  one  of  these 
.sciuimata  with  the  weighting  ol'  the  sccmid  period,  which  is  al- 
ways uneven  in  the  first  and  third  examples,  we  always  get  in- 
dication of  constancy  ;  that  if  we  use  the  geometric  average  upon 


COINCIDKNCK    OF    TFII-:    AVi:i!A(iKS  .'Mil 

evci'v  sin<;lc  (Hie  of  these  scheiunta  witli  the  wci«:htiiiu-  of  holli 
the  [x'l'iods  (oc'ometrieally  eah'iihited,  aeeordiui;'  to  the  S((iiai'e 
roots  of  tlie  pnxhicts  of  the  total  inoiiey-values  at  eacli  period, 
as  explained  in  (Chapter  TV.  Section  V^.  §  <)),  we  always  jrct,  at 
least  very  nearly,  indication  of  constaiK^y. 

1^^-oni  tliese  partieidai"  facts,  the  Hrst  hatch  of  which  lia\"e  al- 
ready been  univei'salized,  arc  to  be  derived  two  ini]»ortant  infer- 
ences, the  Hrst  of  which  can  l)c  directly  nniversalized,  and  the 
second  will  he  nniversalized  ])i'esently  so  far  as  it  admits  ot'  nni- 
versalization. 

§  2.  The  first  of  these  is  this  : — Constant  raass-cpiantities  be- 
ing pnrehase<l  at  each  period,  we  can  make  use  of  the  argnment 
from  compensation  by  e((nal  sums  in  favor  of  the  (trlf/micfic 
average  only  if  we  confine  ourselves  to  using  this  average  with 
the  weighting  of  the  /zr.s/  period  ;  we  can  make  use  of  this  same 
argument  in  favor  of  the  /i(ir)iionic  average  if  we  confine  onr- 
selves  to  using  it  with  the  weighting  of  tlie  secoiuJ  j)eriod  ;  we 
can  make  use  of  the  same  argument  in  favor  of  the  gt'diiirtric 
average  (with  a  modific^ation)  if  we  use  it  with  the  weighting  of 
both  the  periods  together.  The  argument  is  as  good  for  one  of 
the  averages  as  for  another — un(|ualifiedly  in  the  case  of  the  first 
two, — provided  each  be  used  with  a  s})ecial  weighting;  and  for 
them  with  other  weightings  the  argument  has  no  force  whatever. 

Thus  we  see  that  on  the  assum))tion  that  this  argument  is 
correct,  and  on  the  supj)osition  that  it  has  ai)plication  (that  con- 
stant mass-quantities  arc  purchased  at  each  period),  each  of  the 
three  averages  has  for  its  own  jiroper  weighting  each  of  the  three 
kinds  of  weighting  possible — on  the  supposition  of  constant 
mass-quantities  and  varying  total  money-values  of  tlu-  classes. 
In  these  cases  tlie  proper  ii'ei(/htin</  to  use  inif/i  the  (iritlnnetir  (ircr- 
U(/e  of  price  ritriatlonx  ix  the  irei(/htin</  oj  the  first  period  :  the 
proper  loe'u/htiiHf  to  Mne  with  the  hariaonie  (werage  of  price  rarid- 
tions  i.H  the  weiylding  of  the  second  period  ;  the  proper  weighting 
to  use  with  tlie  geometric  average  of  price  variations  is  the  weight- 
ing of  l)oth  the  periods  (the  weight  for  every  class  being  the 
geometric  mean  between  its  two  weights,  one  at  each  period). 

Here  we  mitrht  be  inclined  to  aroue  in  favor  of  the  y;eometric 


.'{oO  THi:    MKTIIOI)    FOR    CONSTANT    MASSES 

average  on  the  ii"i'<>ii'i'l  t'lat  the  weighting  it  re(iiiires  is  alone 
the  right  one.  But  we  ar(>  deterred  from  so  doing  by  the  second 
infercuce. 

§  ."3.  This  second  inference  is  that  all  three  aiiei'age.s,  aich  with 
it'i  01011  proper  iccif/hfinr/,  applied  to  the  same  cmms  with  constant 
ma)<x-(jii<iiititief<  somctuiWK  i/ie/d  the  .same  re,'^ults,  and  <ihrai/s  the 
first  two  i/ield  the  same  results,  and  i)i  mo.^t  (^ordhiai-i/)  cases  the 
third  departs  hut  sllf/Jitli/  from  them. 

That,  when  the  niass-(|nantities  are  constant,  the  arithmetic 
average  of  the  price  variations  with  weighting  according  to  the 
sizes  of  the  classes  at  the  first  period  and  the  harmonic  average 
f)f  them  witli  weighting  according  to  the  sizes  of  the  chisses  at 
the  second  {)eriod  are  identically  the  same  universally  is  demon- 
strated in  Appendix  A,  lY.  ^  .'i.  For  they  both,  being  averages 
of  price  variations,  reduce  to  a  comparison  of  the  same  price 
avei'ages  (arithmetic,  therefore  in  the  same  ratio  as  the  price 
totals).      Tims,  c(tndensing  that  demonstration,  we  have 

p;=.^...U (-^-^l- )    ^'^ 


This  is  nothing  else  than  Serope's  method  of  measuring  varia- 
tions in  the  level  of  prices,  rightly  confined  to  cases  when  con- 
stant mass-cjuantities  are  supposed  to  be  given.  A  shorter,  but 
less  pers])icnons,  formula  for  this  method  is 

P,      a,  +  b,  + ^^> 

Thus  in  all  the  selieniala  em[)loycd  in  this  Chapter,  to  find  the 
constancy  or  variation  of  the  general  level  or  average  of  prices 
(or  the  average  variation  of  prices),  we  have  only  to  divide  the 
total  sum  supposed  t<»  be  spent  on  all  the  classes  at  the   second 


COINCIIUONCK    OF    TIIK    A  \"  i;i!A<i  IvS  .]')  ] 

|)('ri(t(l  \)y  the  totnl  suin  spent  on  tliciii  ;it  the  first  |)cri(>(l.  The 
more  lahoriuiis  ojxTatluiis  of"  averatiiug'  the  price  variations  aritli- 
nietieally  with  the  \vei<;hts  of  the  first  jx'riod,  or  of  uveraj^ing 
tho^ni  harmonically  with  the  weights  of  thcsec(mcl  period,  may  be 
dispensed  with. 

That,  on  the  same  assumj)tioii  of  constan(;y  in  the  mass-quan- 
tities purciiascd  at  both  periods,  with  additional  confinement  to 
the  su])position  that  we  are  dealing  with  the  variations  of  only 
tiro  classes,  which,  furthermore,  are  such  that  the  geometric 
means  of  their  two  weights  (at  the  two  ])eriods)  are  equal,  so 
that  we  can  use  the  geometric  average  of  their  price  variations 
with  even  weiy-hting,  the  geometric  averaw  \vith  this  weiychtino- 

a  c5/  »^  «r>  oft/ 

that  is,  the  geometric  mean,  universally  yields  exactly  the  same 
results  as  the  other  two  averages,  a])})lied  to  these  same  cases, 
each  with  its  proper  weighting,  is  demonstrated  in  Appendix  A, 
VI.  §  7.  There  also,  in  §  9,  it  is  shown  that  in  all  cases,  with 
any  number  of  classes,  with  variously  nneven  weighting,  with 
all  the  moderate  irregularities  in  the  sizes  of  the  classes  such  as 
are  likely  to  exist  in  our  statistical  lists,  the  geometric  average 
with  its  proper  w^eighting  always  yields  results  very  nearly  the 
same  as  the  single  results  yielded  by  the  other  two  averages, 
each  with  its  proper  weighting.  Of  these  more  complex  cases 
we  shall  treat  in  a  later  Section.  Here  we  may  continue  to  con- 
fin(»  our  attention  mostly  to  the  simple  cases  where  all  three  of 
the  averages,  oi-  rather  means,  each  with  its  proper  weighting, 
coincide. 

§  4.  A  pretty  illustration  of  this  coincidence  may  be  offered 
in  the  following,  which  provides  us  wnth  what  may  be  called  an 
argument  by  transposition  of  pric(>s.  Suppose  we  have  at  each 
period  1  A  and  1  B,  and  it  happens  that  their  prices  become 
transj)osed,  that  is,  that  the  price  of  A  at  the  second  [)eriod  is 
the  same  as  B's  at  the  first  and  that  of  B  at  the  second  the  same 
as  A's  at  the  first,  so  that  a^  =  j-l^  and  j^.^  =  a^.  Then  also  the 
importance  of  these  articles  is  trans|)osed — and  if  there  be  the 
same  number  of  them  in  their  classes,  the  sizes  of  the  classes  are 
to  each  other  as  these  prices,  and  are  transposed  (as  in  the  third 
schema  in  the  last  set  above).     Under  these  variations  it  is  per- 


'S'rl  TIIK    .MKTIIOD    FOl!    CONSTANT    MASSES 

fectly  evident  that  the  exchan^e-vahie  of  nutney  suffers  im  varia- 
tion. Now  this  eonstaney  of  the  exchan*»;c-value  of  money,  as 
shown  ill  the  level  t)f  prices,  is  indicated  by  the  arithmetic 
avera»;-e  of  these  price  variations  with  the  weitrhtinii'  of  the  first 

l)eriod,  thus ' ',  and  1)\'  tlie  iiai'nionic  averau'e  of   thetn 

^^,  +  .^.  ■ 

Avith  tlie  \\ei<>-htinu'  of  tlie  second  period,  thus '  "  -  ,  he- 

a,  +  ,i,  . 

cause  each  of  these  reduces  to    '       '.",  wliich,  on  account  of  tlie 

"^  +  ,^. 
equality  between  a.^  and  ,9,  and  between  fi.-^  and  a^,  is  equal  to 

1.00  ;  and  also  by  the  geometric  average  Avith  the  weighting  of 
both  the  periods,  which  is  even,  since  n/«j«.,  =  v/y?,;?.,,  because 

the  exi)ressi()n  A—--^  is  ecaial  to   1.00  also  (in  account  of  the 
\^/,    ,i, 

equality  between  those  terms.  Henc(>  the  argument  based  uj)on 
this  jierfect  evidence  is  equally  good  for  either  of  these  averages, 
each  with  its  own  ])roper  weighting.' 

§  5.  Oil  account  of  this  coincidence,  when  the  physical  foun- 
dations are  the  same  at  both  jieriods,  it  is  futile  to  argue  for  any 
one  average  alone,  or  for  any  one  weighting  alone,  in  preference 
to  anv  other.  As  regards  mere  convenience,  Scrope's  method  is 
the  most  serviceable  of  all,  in  these  cases  ;  and  this  represents 
no  one  of  the  three  averages  of  the  jirice  variations  more  than 

'  'Hiis  cxiiiiiplc  may  1)0  widened,  sliowiiit;  the  s:mie  identity  between  the  aver- 
ages dilt'erently  weinlited.      Let  tlie  point  O,  soniewliere  divide  the  line  A  I!  at  the 
A (),  (),  B 

first  period,  and  the  point  O.,  somewhere  divide  it  at  the  second  period.  Then  the 
segment  .VOj  has  become  AO2,  and  the  segment  OiB  has  become  OnV>,  but 
these  variations  of  tiie  segments  have  not  affected  the  whole  line  ;  wherefore  the 
average  of  the  variations  evicU-ntly  is  unity,  indicating  constancy.  Now  what- 
ever be  the  variations,  provided  neither  segment  be  zero  in  length  at  either 
period,  constancy  is  shown  by  the  arithmetic  average  of  them  with  the  weight- 
ing of  the  first  period,  and  by  the  harmonic  average  of  them  with  the  weight- 
ing of  the  second  period;  and  provided  tliey  be  such  that  AOj  ^  O.^H  and 
A02=0]B,  also  the  geometric  mean  (with  even  weighting)  indicates  constancy 
(and  in  other  cases  the  geometric  averages  with  the,  weighting  of  both  ])eri()ds, 
provided  the  weight  of  the  one  be  not  more  than  two  or  three  times  larger  than 
that  of  the  other,  gives  a  result  very  nearly  e(|nal  to  unity). 


COINCIDENCE    OF    THE    AVERACJEK  353 

another,  except  in  complex  cases,  when  the  geometric  average 
deviates.  There  is  little  to  say  in  favor  of  the  arithmetic  aver- 
age, or-  in  fevor  of  the  weighting  of  the  first  period,  each  by 
itself;  bnt  if  together  they  give  the  right  result,  we  cannot  com- 
plain. Similarly  with  the  harmonic  average  and  the  weighting 
of  the  second  period.  The  average  and  the  weighting  that  have 
most  reason  in  their  favor  singly  are  the  geometric  and  the 
weighting  of  both  the  periods.  Yet  we  find  that  these  give  no 
better  results  than  the  others  properly  combined, — and  in  com- 
plex cases  we  shall  find  that  their  results  are  not  so  good  as 
those  of  the  others.  The  greatest  errors  arise  from  using  an 
average  with  weighting  not  suitable  to  it — the  arithmetic  with 
the  weighting  of  the  second  or  of  both  periods,  the  harmonic 
with  the  weighting  of  the  first  or  of  both  periods,  the  geometric 
with  the  weighting  either  of  the  first  or  of  the  second  period 
alone. 

In  reality,  however,  not  one,  but  all  the  averages  and  weight- 
ings are  used.  In  Chapter  IV.  (Sec.  V.  §  4)  we  found  it 
absurd  to  use  the  weighting  of  either  period  alone.  But  now, 
in  using  Scrope's  method,  we  are  not  using  the  weighting  of 
either  period  alone.  For  we  are  using  the  weighting  of  the  first 
period  alone  only  with  the  arithmetic  average  ;  but  equally  well 
are  we  using  the  weighting  of  the  second  period — alone  with  the 
harmonic  average.  The  one  true  method  which  combines  both 
these  averages,  combines  both  these  weightings.^  And  in  the 
special  cases  where  the  result  given  by  it  is  necessarily  given 
also  by  the  geometric  mean,  this  is  with  the  use  of  the  combined 
weightings  of  both  the  periods.^ 

2  In  Note  12  in  that  Chapter  and  Section  it  was  stated  that  a  variation  is  not 
properly  a  variation  of  the  individuals  existing  at  either  period  alone,  but  it  is 
from,  the  individuals  at  the  first  period  to  the  individuals  at  the  second.  We  have 
now  found  that  in  averaging /ro?«  individuals  at  the  first  period  the  proper  aver- 
age to  use  is  the  arithmetic  with  the  weighting  of  the  first  period  ;  and  that  in 
averaging  to  individuals  at  the  second  period  the  proper  average  to  use  is  the  har- 
monic with  the  weighting  of  the  second  period.  Thus  we  have  perfect  harmony 
throughout. 

•■'Students  of  German  philosophy  will  notice  a  peculiar  resemblance  between 
the  three  averages  and  the  three  categories  in  each  of  Kant's  four  divisions,  and 
the  three  terms  in  the  trichotomy  of  Hegel.  For  the  arithmetic  and  harmonic 
averages  are  opposed  to  each  other  (although  with  likewise  opposite  weighting 

23 


354  THE    .MK'IIIOI)    FOR    COXSTAXT    MA.SSRS 

§  6.  Consequently  in  liis  dispute  with  Laspeyres,  in  which 
the  price  of  one  article  was  supposed  to  rise  from  1.00  to  2.00 
and  that  of  another  to  fall  from  1.00  to  .50,  Jevons  would  have 
been  right  in  considering  the  exchange-value  of  money  con- 
stant, as  indicated  by  the  geometric  average  with  even  weight- 
ing, if  he  could  have  shown,  or  had  added  as  part  of  the  sup- 
position, that  the  two  classes  were  equally  important  over  both 
the  periods  together.  But  in  this  case  the  other  two  averages, 
cavh  with  its  proper  weighting,  would  make  the  same  indication. 
On  the  other  hand  Laspeyres  would  have  been  right  in  using  the 
arithmetic  average  with  even  weighting,  and  in  concluding  that 
the  price  level  had  risen  by  25  ])er  cent.,  had  he  been  careful  to 
specify  that  he  was  dealing  with  classes  equally  important  at 
the  first  ])eriod,  and  only  with  such.  But  in  this  case  the  har- 
monic av(>rage  with  its  pro]>er  weighting  would  make  the  same 
indication,  while  the  geometric  average  with  its  proper  weight- 
ing, would  diverge  only  by  indicating  a  rise  of  2(5  jkh-  cent. 
Again,  Jevons,  in  suggesting  the  use  of  the  harmonic  average, 
still  with  even  weighting,  which  would  indicate  a  fall  by  20  per 
cent.,  would  have  been  right,  iiad  he  rested  on  the  condition  of 
the  two  classes  being  ecpially  important  at  the  second  period 
only.  But  in  this  case  also  the  arithmetic  average  with  its 
proper  weighting  would  make  the  same  indication,  and  the 
geometric  average  with  its  projier  weighting  would  diverge  only 
by  indicatmg  a  fall  of  20.(34  per  cent.  All  this,  however,  is 
said  only  on  the  suj^position  that  both  these  writers  agreed  in 
assuming  that  constant  mass-quantities  of  each  class  were  pur- 
chased at  each  j)erio<l.^ 

they  agree),  and  the  geometric  aversige synthesizes  these  opposites  (with  weighting 
which  likewise  synthesizes  their  weightings,  provided  this  composite  weighting 
be  even).  Unfortunately  tiiis  is  exactly  so  only  when  the  geometric  average  is 
restricted  to  being  a  mean  proper. 

•*  This  assumption  was  used  in  Laspeyres's  reasoning,  as  also  that  the  classes 
were  equally  important  at  the  first  period,  and  so  he  happened  to  be  right  in  his 
conclusion,  confined  to  these  conditions.  But  he  never  recognized  these  condi- 
tions, nor  confined  his  argument  to  them,  so  that  in  general  his  position  was  no 
better  than  Jevons's.  Had  .Jevons  made  the  assumption  that  const.iint  equal  sums 
were  spent  on  the  two  classes,  his  choice  of  the  geometric  mean  would  have  been 
exactly  right,  as  we  have  seen, — but  not  so  his  choice  of  the  geometric  average  in 
wilier  cases.  Hut  he  made  tiiis  assumption  only  in  connection  with  tli'e  harmonic 
average,  which  ilicn  is  wrong. 


SCHOI'K's    MKIIIOD  355 

On  this  assumption  we  tintl  Jevons's  pr()[)liecy  fulfilled.  Ht" 
supposed  another  case,  in  which  the  price  of  one  article  remains 
unchanged  at  1.00  and  that  of  another  i-iscs  from  1.00  to  2.00, 
and  said  that  "  the  mean  rise  of  price  mio;ht  be  variously  stated  " 
as  the  arithmetic  at  50  per  cent.,  as  the  geometric  at  41,  or  as  the 
harmonic  at  33,  and  added  the  sentence  already  quoted  :  "  It  is 
probable  that  each  of  these  is  right  for  its  own  purposes  when 
these  are  more  clearly  understood  in  theory."  ^  Strictly  speak- 
ing, none  of  the  averages  has  any  purposes  of  its  own,  but  each 
one  is  subject  to  certain  conditions.  Thus  in  the  case  supposed, 
with  further  assumption  of  the  mass-quantities  being  constant, 
the  arithmetic  mean,  indicating  a  rise  of  50  per  cent.,  is  right  if 
the  two  articles  were  equally  important  at  the  first  period,  the 
harmonic,  with  rise  by  33  per  cent.,  if  they  were  equally  impor- 
tant at  the  second  period,  and  the  geometric,  with  rise  by  41  per 
cent.,  if  they  were  equally  important  over  both  the  periods.  If 
their  importance  was  wholly  uneven,  the  mean  rise  might  be  auv 
figure  between  0  and  100  per  cent. — and  the  right  figure  would 
be  indicated  either  by  the  arithmetic  average  with  the  weighting 
of  the  first  period,  or  by  the  harmonic  average  with  the  weight- 
ing of  the  second  period,  or  (approximately  in  many  cases)  by 
the  geometric  average  with  the  weighting  of  both  the  jieriods." 
But  we  have  anticipated  somewhat,  and  must  now  seek  to 
prove  that  this  common  result  of  the  two  averages  always,  and 
sometimes  of  all  three,  is  the  right  one. 


III. 

§  1.  The  reason  why  the  argument  from  compensation  by 
equal  sums  has  been  mistaken  for  an  argument  specially  favoring 
the  arithmetic  average  of  price  variations  is  because  in  making 
use  of  it  the  advocates  of  this  average  have  had  in  mind  such 
conditions  as  are  illustrated  in  these  three  schemata, — for  the 
arithmetic  price  variations  : 

5B.  23,  p.  121. 

*  We  have  seen  something  similar  in  the  preceding  Chapter.  But  there  the 
weightings  corresponding  to  these  were  of  another  kind,  and  hidden  (except  in 
the  tliird  instance),  so  as  not  to  be  serviceable  (being  there,  as  here,  inexact  in 
the  third  instance). 


356 


THE    MP:TH()r)    FOK    CONSTANT    MASSES 


I    100  A  @1. 00     100  n@  1.00        100  for  [A] 
II    100  A  @  1.50     100  B@    .50       150  for  [A] 

for  tlie  harnionio  price  variations: 

I    100  A  @  1.00     100  B  (n\  1.00     I  100  for  [A] 
II    100  A  ((V,  1..50     100  B  Or,    .75     |  1-50  for  [A] 

for  the  geometric  price  variations  : 


100  for  [B]  —  200, 
50  for  [B]  —  200; 


100  for  [B]— 200, 
75  for  [B]— 225; 


I   100  A  @,  1.00     100  B  @  1.00 
II   100  A  @  1.50    100  B  @    .661 


100  for  [A]     100    for  [B]  —  200, 
1.50  for  [A]       m'i  for  [B]  —  216s; 


in  which,  the  classes  always  being  supposed  equally  large  at  the 
first  period,  compensation  by  equal  sums  (and  by  equal  prices) 
takes  place  only  in  the  first  example,  where  also  the  arithmetic 
average  with  even  weighting  indicates  constancy.  If,  however, 
any  (me  had  wanted  to  use  this  argument  for  the  harmonic  aver- 
age of  price  variations,  he  might  have  made  use  of  the  following 
schemata, — for  the  arithmetic  price  variations  : 

I    100  A  @  1.00     100  B^  @  3.00 
II    100  A  Or,  1.50     100  B^  @  1.50 

for  the  harmonic  price  variations 


100  for  [A]     300  for  [B]  —  400, 
150  for  [A]     150  for  [B]  —  300; 


I   100  A 
II   100  A 


1.00     100  B'@  2.00 
1.50     100  B^®  1.50 


100  for  [A]     200  for  [B]  —  300, 
150  for  [A]     150  for  [B]  —  300; 


for  the  geometric  price  variations  : 

I    100  A  @  1.00     100  B^  @  2.25  I  100  for  [A]     225  for  [B]  —  325, 
II    100  A  @  1.50     100  B'  (oj  1..50  |  150  for  [A]     L^O'for  [B]  —  300; 

in  which,  the  classes  always  being  supposed  equally  large  at  the 
second  period,  com])ensation  by  equal  sums  (and  by  equal  j)rices) 
takes  place  ouly  in  the  second  example,  where  constancy  is  in- 
dicated, even  weighting  being  used,  only  by  the  harmonic  aver- 
age. Agaiu,  if  any  one  had  wanted  to  use  this  argument  for 
the  geometric  mean  of  price  variations,  he  might  have  made  use 
of  these  schemata, — for  the  arithmetic  price  variations  : 

100  for  [A]     173.20  for  [li]  —  273.20, 
150  for  [A]       86.60  for  [B]  —  236.60; 


I   100  A  @  1.00     100  B^''@  1.7320 
II   100  A  @  1.50     lOOB'''®    .8660 

for  the  harmonic  price  variations 


I    100  A  @  1.00     100  ir'@  1.4142 
II    100  A  @  1.50     100  ir  @  1.0606 


100  for  [A]     141.42  for  [B]  —  241.42, 
150  for  [A]      106.06  for  [B]  —  256.06; 


for  the  geometric  price  variation^ 


.schoi'k's  m];t[I()I)  -  .357 


I    100  A  @  1.00     100  B''  @  1.50 
II    100  A  0,\  1.50     100  B''  @  1.00 


100  for  [A]     150  for  [B]  —  250, 
150  for  [A]     100  for  [B]  —  250; 


in  which,  tlie  classes  always  being  supposed  equally  large  over 
both  the  periods  together,  compensation  by  ecpial  sums  (and  by 
equal  prices)  takes  place  only  in  the  third  example,  where  con- 
stancy is  indicated,  even  weighting  being  used,  only  by  the 
geometric  average. 

The  error  in  all  these  applications  of  the  argument  is  mere 
ignoring  of  the  fact  that  each  of  the  averages  with  its  own 
proper  weighting  applied  to  every  one  of  these  examples  gives 
exactly  the  same  results  (except  in  those  where  the  weighting 
for  the  geometric  average  is  not  even,  its  results  then  deviating 
somewhat). 

§  2.  But  although  we  cannot  use  the  argument  to  distinguish 
between  the  three  averages,  we  can  use  it  to  show  the  correct- 
ness of  all  three  averages  (the  third,  however,  only  in  special 
cases),  each  with  its  own  proper  weighting — always  on  condition 
of  constant  mass-quantities.  If  we  bring  together  the  three 
simplest  schemata  on  which  the  advocates  of  each  average  may 
rely,  as  follows — for  the  arithmetic  price  variations  : 


I   100  A  @  1.00    100  B  @  1.00 
II   100  A  @  1.50     100  B@    .50 

for  the  harmonic  price  variations 

I   100  A  @  1.00     100  B'  @  2.00 
II    100  A  OJ)  1.50     100  B^  @  1.50 


100  for  [A]     100  for  [B]  —  200, 
150  for  [A]       50  for  [B]  —  200; 


100  for  [A]     200  for  [B]  —  300, 
150  for  [A]     150  for  [B]  —  300; 


for  the  geometric  price  variations 


I   100  A  @  1.00     100  B''  @  1.50 
II   100  A  @  1.50     100  B'^  @  1.00 


100  for  [A]     150  for  [B]  —  250, 
150  for  [A]     100  for  [B]  —  250; 


we  see  the  reason  for  the  correctness  of  each  average  with  its 
own  proper  weighting — always  indicating  constancy  with  perfect 
clearness  only  in  the  third  example  ;  for  in  the  others,  although 
we  have  compensation  by  equal  sums,  it  is  on  classes,  and  indi- 
viduals in  them,  that  have  different  exchange-values  over  both 
the  periods  together.  But,  as  already  noticed,  there  is  a  cor- 
rection accompanying  each  deviation.  In  the  first  example,  the 
class  [B]  is  less  important  than  the  class  [A]  over  both  the 


:]oS  TlIK    METHOD    FOR    CONSTANT    MASSES 

periods  together  ;  hut  its  price  falls  more  than  the  price  of  [A] 
rises,  since  we  know  that  a  fall  from  1.00  to  .50  is  greater  than 
a  rise  from  1.00  to  l.oO.  And  in  the  second  case  the  class  [B] 
is  more  important  than  the  class  [A]  over  hoth  the  periods  to- 
gether ;  but  its  price  falls  less  than  the  price  of  [A]  rises,  since 
we  know  that  a  fall  from  1.00  to  .75  is  smaller  than  a  rise  from 
1 .00  to  1 .50.  Still  we  do  not  yet  perceive  that  in  the  former 
case  the  price  of  [B]  falls  exactly  as  much  more  as  it  ought  to 
do,  to  make  up  for  the  smaller  importance  of  its  class  ;  nor  that 
in  the  latter  case,  it  falls  exactly  as  much  less  as  it  ought  to  do, 
to  allow  for  the  greater  importance  of  its  class.  But  we  may 
learn  it  with  perfect  certainty  by  the  following  reasoning. 

The  first  example  may  be  converted  into  the  following,  al- 
ready used,  by  merely  employing  a  different  mass-unit  of  [B] , 
Avith  prices  inversely  altered  : 


I   100  A  @  1.00    57.73  B''  @  1.7320 
II    100  A  @  1.50     57.73  B''  @    .8660 


100  for  [A]     100  for  [B]  —  200, 
150  for  [A]       50  for  [P>]  —  200. 


Here  the  mass-unit  of  [B]  is  equivalent  to  the  mass-unit  of 
[A]  over  both  the  periods,  so  that  A  and  B"  may  be  taken  as 
economic  individuals.  Now  we  purchase  157.73  such  indi- 
viduals at  each  period,  and  we  pay  exactly  the  same  sum  for 
them  at  each  ])eriod.  Therefore  our  money  has  retained  exactly 
the  same  purchasing  power  or  exchange-value  over  both  the 
periods.^     And  the  second  example  may  be  converted  into  the 

1  The  fact  that  1(M»  of  these  iiulividuals  are  in  [A]   and  .")7.7;>  in   [B]  merely 
shows  that   [A]   is  ^---  =^  1.732  times  hirger  than  [B].     This  we  already  know 

from   tlic  fact   that  A/TTlTrS  m^  == ''^=  ^•"'^^"''•'-     ^^''^''   "''''  weighting  tlie  geo- 

metric  average  indicates  a  rise  by  0.3  per  cent.,  and  is  by  so  much  wrong. — In 
C'hapt.  VIII.  Sect.  III.  ?1  we  saw  inconsistency  in  the  argument  of  the  arithmetic 
averagist  on  the  supposition  of  the  classes  always  being  e(|ually  important.  But 
here  the  two  classes,  equally  important  at  the  commencement,  are  not  thereafter 
equally  important,  and  the  inconsistency  vanishes.  Tluis  the  arithmetic  averagist 
argues  that  if  [A]  and  [B]  are  equally  important  at  the  first  period  their  com- 
pensatory variations  should  be  to  equal  distances  from  their  equal  starting  points. 
Suppose  A  rises  from  1.00  to  L.TO  and  B  falls  from  1.00  to  .50.  Then  [AJ  is  three 
times  more  important  than  [B],  and  therefore,  starting  from  this  position  as  a 
new  iJrst  period,  the  price  of  S  A  should  rise  from  1.00  only  one  third  as  far  as  tiie 
price  of  2  B  falls  from  1.00.  This  is  precisely  what  takes  place  when  A  continues 
to  rise  from  1.50  to  1.51,  while  B  cofttinues  to  fall  from  .50  to  .49. 


scuopk'k  method  l]')^.) 

followintj;,  also  already  used  : 


I    100  A  @  1.00     141.42  B''  @,  1.4142 
II    100  AT")  1.50     141.42  B^'C'^)  l.OdOC) 


100  Cor  [A]     200  for  [B]  —  300, 
ir)0  for  [A]     150  for  [?.]  —  300. 


Here  also  the  niass-nnit  of  [B]  is  e(jaivalcnt  to  the  mass-unit 
of  [A]  over  both  the  [)eriods,  .so  that  A  and  B"  may  be  taken 
as  economic  individuals.  Now  we  purchase  241.42  such  indi- 
N'iduals  at  each  period,  and  we  pay  exactly  the  same  sum  for 
them  at  eac^h  period.  Therefore  our  money  has  retained  ex- 
actly the  same  purchasing  power  or  exchange-value  over  both 
the  periods.^ 

§  3.  We  thus  obtain  also  here — that  is,  applicable  only  to 
cases  with  constant  mass-quantities — a  precise  method  of  meas- 
uring the  dt)nstancy  or  variation  of  the  general  exchange-value 
of  money.  It  is  :  Find  iti  all  the  classes,  for  u.sc  as  units, 
masses  that  have  tJie  sainc  money-value  over  both  the  periods  to- 
gether, and  measure  the  constaticy  or  variation  of  the  exchange- 
vcdue  of  money  inversely  by  the  consfaney  or  variation  i^i  the  toted 
sum  of  money  needed  at  each  period,  to  purchase  the  constant  num- 
bers of  these  mass-unitx  supposed  to  be  actuedly  pmrchased .  Nat- 
urally this  method  is  not  confined  to  cases  with  only  two  cla.sses. 

In  this  method,  however,  the  first  part  is  superfluous,  since, 
whatever  be  the  mass-quantities  in  the  various  classes,  we  know 
that  they  must  contain  certahi  numbers  of  mass-units  that  are 
equivalent  over  both  the  periods,  which  numbers  will  be  con- 
stant if  the  mass-quantities  are  constant ;  but  as  we  make  no 
use  of  these  numbers  when  ascertained,  it  is  unnecessary  to  as- 
certain them.  All  we  need,  then,  is  to  measure  the  constancy  or 
variation  of  the  exchange-value  of  money  inversely  by  the  con- 
stancy or  variation  in  the  toted  sum  of  money  needed,  at  each 
period  to  purchase  the  constant  mass-quantities  of  all  the  classes. 

The  formula  carrying  out  this  method,  we  may  repeat,  in  its 
simplest  form,  is  the  following, 

-Here,  [B]  containing  141.42  and  [A]  100  of  these  individuals,  the  former 
is  1.4142  times  larger  than  the  latter.     This  we  also   know  from  the  fact  that 

V900  .^  1  p;o        -  ♦ 

"^ —  ='2=1.414213.     With  this  weighting  the  geometric  average  indicates 

100    •:   1.50  6  6  6  6 

a  fall  by  0.06  per  cent.,  and  is  by  so  much  wrong. 


.'^()()  THK    M1:TH()T)    FOl!    CONSTANT    .MASSF.S 

P.,     a„  -I-  b.,  +  c„  + 

p;  ==  a; +  b;  +  c; +  .:.-'  ^"^> 

or  this, 

(in  which  it  is  evident  the  sizes  of  the  muss-units  used  have  no 
influence).  The  last,  to  repeat  also,  we  recognize  to  be  the 
formula  for  Scrope's  method,  which,  therefore,  is  the  correct 
method  for  the  cases  in  question. 

Now  in  the  preceding  Chapter  (Sect.  III.  §  .">)  we  found  the 
method  there  discovered  for  cases  with  coustimt  sums  to  be 
Scrope's  method  applied  to  the  geometric  means  of  the  mass- 
quantities.  But  if  we  take  the  formula  for  that  forni  of  Scrope's 
method  (there  given  as  No.  8),  and  apply  it  to  cases  in  which 
the  two  mass-quantities  in  every  class,  the  one  at  the  one  period 
and  the  other  at  the  other,  are  the  same,  the  formula  reduces 
to  the  last  formula  above.  Or,  reversely,  by  distinguishing  x 
into  x^  and  x^,  to  represent  the  mass-quantities  at  each  period 
notwithstanding  that  they  are  equal,  and  distinguishing  the 
other  symbols  for  the  mass  quantities  in  the  same  way,  we  may 
derive  from  the  last  expression  this, 

&  =  «2^-y2  +  ^yvxyz  +  r-yhh  + /g^ 

Pi    ayx^x^  +  i^yy^2  +  ry  V2  + ' 

Thus  Scrope's  method  applied  to  the  geometric  means  of  the  mass- 
quantities  is  a  comprehensive  method,  applying  both  to  the  cases 
with  constant  sums  of  money  and  to  the  cases  ivith  constant  mass- 
quantities. 

§  4.  Therefore,  like  the  method  examined  in  the  preceding 
Chapter  for  the  cases  with  constant  sums,  this  method  for  the 
cases  with  constant  mass-quantities  satisfies  all  the  Propositions 
that  more  or  less  definitely  prescribe  what  the  variation  of 
money  in  exchange-value  in  all  other  things,  and  consequently 
the  inverse  variation  of  tlu;  general  level  of  prices,  must  be. 
Thus  we  see  plainly  that  it  indicates  constancy  if  no  prices  vary, 
and  if  all  prices  vary  alike  it  indicates  the  same  variation  (Propo- 


kcrope's  ^[ethoi)  ;361 

sitions  XXVII.,  XLI V.,  XVII.,  XLV.),  no  matter  what  be 
the  weights  (provided,  of  course,  they  be  such  that,  with  the 
price  variations,  they  show  the  mass-quantities  to  be  constant) ; 
nor  can  it  indicate  constancy  if  there  is  only  one  price  variation, 
or  if  all  price  variations  are  in  one  direction — and  in  the  latter 
event  it  of  course  cannot  indicate  a  variation  in  the  opposite 
direction  (Propositions  XX.  and  XXVIII.).  It  also  satisfies 
Proposition  XXXVI.,  if  its  own  condition  be  observed  in  the 
omitted  classes.  This  is  so  plain  as  not  to  need  to  be  demon- 
strated. But  again  a  similar  remark  has  to  be  made  here  as 
was  made  in  the  corresponding  passage  in  the  preceding  Chapter 
(Sect.  III.  §  4).  That  Proposition-^  does  not  require  the  omitted 
class  or  classes  to  be  of  the  same  mass-quantities  at  both  the 
periods.  Yet  this  method  requires  that  they  should  be  ;  for  if 
they  are  not  and  its  principal  formula,  (3),  is  extended  to  them 
(but  altered,  so  as  to  distinguish  between  the  new  mass-quanti- 
ties at  each  period),  the  two  results  will  not  agree.  Yet  it  is 
not  this  method  which  is  being  used  and  disproved,  but  another 
method ;  and  this  method  satisfies  the  test  oifered  by  that  Propo- 
sition perfectly.'' 

§  5.  Moreover,  as  above  seen,  the  arithmetic  average  of  the 
price  variations  with  the  weightmg  of  the  first  period  and  the 
harmonic  average  of  them  with  the  weighting  of  the  second 
period  are,  under  the  given  condition,  universally  the  same  as 
this  method.  Therefore  these  averages,  with  these  weightings, 
are  both  correct.  But  they  are  superfluous,  since  Scrope's 
method  is  simpler  and  more  convenient. 

And  the  geometric  average  with  the  weighting  of  both  periods 
also  reduces  to  this  method  in  all  cases  when  we  are  dealing  with 
the  variations  of  two  equally  (over  both  periods)  important 
classes,  so  as  to  be  able  to  use  even  w^eighting.  This  is  so  not 
only  when  the  result  indicated  is  constancy,  but  also  when  the 
result  indicated  is  a  variation.     We  have  already  noticed  several 

^  Nor  Proposition  XXXII.,  when  the  prices  are  constant. 

*  But  again  a  similar  qualification  has  to  be  added  here  also.  What  is  said  in 
the  text  may  be  said  with  reference  to  formula)  either  (1),  (2),  (3),  or  (4).  But 
formula  (5)  satisfies  the  Proposition.  In  other  words,  this  method  satisfies  the 
Proposition  if  we  treat  the  additional  different  mass-quantities  as  this  formula 
prescribes. 


362  TiiK  mi:tii()1)  for  constant  masses 

instances  of  tlic^  former  kind  :   we   may  now  notice  a  eonj)le  of 

the  latter.      Tliese  may  he  taken  from  the  hist  set  of  schemata 

given  ahove  in  ij  1,  where  the  hihor  lias  already  been  performed 

of  obtaining  not  only  classes,  but  individuals,  equally  important 

over  both  periods.     In  the  first  of  these  examples  we  see,  there- 

P        236.6 
fore,   that   we   have  -p^  =  Q^..:^?)  =  0.8660,  indicating  a  fall  of 

13.40  per  cent.      The  geometric  mean  of  the  price  variations  is 

3       1       v/3 

X  .^  =    .,    =  0.H660,    indicating   the   same    fall.       In    the 


J 


P.,      256.0()  .    ^.      . 

second  example  we   have  p"  =  ;;—      -^  =  l.OiKXi,  mdicatmg  a 

|3     ^         3 

rise  of  ($.06  i)(>r  cent.      The  s'cometric  mean  is  .        x       =  

'  N2        4        %/2 

=  1.060G,  indicating  the  same  rise.    • 

Tn  each  of  these  examples  it  may  be  noticed  that  the  vari- 
ation of  general  prices  is  from  1.00  to  the  same  figure  as  in  the 
price  of  B"  at  the  second  period.  This  relation  is  universal 
under  the  conditions  supposed.  These  are  that  «j"=1.00, 
x"  =  y"  and  a,a„  =  b,b.,.      For  from  the  latter,  with  the  aid  of 

T    .      a,      b.,     a.,  -(-  b.,     x"o," 

other  known   relations,  we  derive  ~  =  —  =     "  '  =     ,— -— = 

b,      a,      a,  +  b,      //",?/' 

"«/'~.r"«/'  +  y"^/'-F~  Mo~  ri/'~  '^/'~'  -    ~  «/'  +  /V' 


X 


J 


'^■i\     Q.  E.  D.^ 


With  uneven  weighting  the  geometric  average  of  the  price 
variations  does  not  necessarily  agree  with  the  common  result 
given  by  the  other  two  averages,  each  with  its  proper  weighting. 
In  showing  that  the  other  two  averages  always  agree  with  the 
])roper  method  we  have  shown  that  their  common  result  is  right. 
AYe  need,  therefore,  not  so  much  to  show  that  the  geometric 
average  is  wrong  when  it  diverges,  as  to  investigate  its  devi- 
ations.     It   is  evident   at   once  that  the  geometric  average  will 

*  Compare  these  with  correspoiidiiig  rchitions  in  the  preceding  Chapter,  Sect. 
II.  ?4,  Note.'?. 


DKVIATIOX    OF    THE    GEOMETRIC    AVERAGE  36."j 

agree  with  the  true  method  here  in  all  cases  correspouding  to 
the  cases  in  whicli  we  have  found  that  it  would  agree  with  the 
other  true  method,  in  the  preceding  Chapter  (Sect.  III.  §  5). 
Some  inferences  similar  to  inferences  in  that  chapter  made  about 
the  deviations  will  also  follow. 

IV. 

§  1.  What  we  desire  to  prove  and  to  illustrate  may  be  shown 
here  on  a  single  example.  Let  us  suppose  the  class  [B]  to  be 
twice  as  important,  over  both  the  periods  together,  as  the  class 
[A].  The  price  of  A  rising  from  1.00  to  l.oO,  to  have  con- 
stancy according  to  the  geometric  average  Avith  this  weighting, 
the  price  of  B  must  fall  from  1.00  to  \/|  =  .8165  ;  and  in  order 
to  have  this  weighting  with  these  price  variations,  the  mass- 
quantities  must  be  in  the  following  proportions  : 

I  100  A  @  1.00    271.08  B@  1.00      MOO  for  [A]    271.08  for  [B] —371.08, 
II  100  A  @  1.50    271.08  B@    .8165  1 150  for  [A]    221.34  for  [B] —371.34, 


in  which  s/lOO  x  150  =  I  n/271.08  x  221.34.     Here  Scrope's 

371.34 
method   shows   a  rise  of   prices,   viz.,   ^w^  ^.^  =  1.000G98 — a 

rise  by  0.0698  per  cent.     That  this  is  right  is  more  apparent 
upon  rearranging  the  mass-unit  of  [B]  as  follows  : 


I  100  A  @  1.00     200  B'^  @  1.3554 
II  100  A  @  1.50     200  B''  @  1.1067 


100  for  [A]      271.08  for  [B]  —  371.08, 
100  for  [A]      221.34  for  [B]  —  371.. '54. 


For  here  the  mass-unit  of  [B]  is  of  the  same  exchange-value 

over  both  the  periods  as  the  mass-unit  of  [A] ;  wherefore,  as  we 

have  to  pay  more  for  these  equivalent  mass-units  at  the  second 

period,  it  is  evident  tliat  their  prices  have  risen.     Thus  Scrope's 

method  is  right  in  its  indication,  and  the  geometric  method  errs 

by  indicating  constancy  when  it  ought  to  indicate  a  slight  rise, 

and  so,  in  this  case,  it  is  slightly  below  the  truth  (by  0.06976 

per  cent.). 

In  this  example  the  figures  for  [B]  could  also  be  arranged  as 

follows  : 

I  180.72  Bi@  1.50, 
II  180.72  Bi@  1.2247, 
or 


364  'I'lii-:  MKiHoi)   I'oi;  ( onsi-ant   massks 

I  221.34  B"  @  1.2247, 
II  221.34  B"  @  1.00, 

these  falls  being  the  same  as  from  1,00  to  .^l()0,  and  the  sums 
spent  on  [B]  being  the  same  as  before.  Or  we  could  take  half 
of  each  of  these,  as  follows  : 

I  90.36  Bi  @  1.50     +  110.67  B"@  1.2247, 
II  90.36  B'  @  1.2247  +  110.67  B"  @  1.00  ; 

and  here  the  price  of  [B]  has  fallen  from  l.")0  to  1.00,  in- 
versely as  the  price  of  A  has  risen  from  1.00  to  1.50.  But  it 
is  not  equal  masses  of  [B]  that  hand  on  this  fall.  We  virtu- 
ally have  three  classes.  And  in  the  quantities  indicated  the 
three  classes  are  equally  important  over  both  the  periods  ;  and 
the  geometric  average  and  the  true  method  give  the  same  diver- 
gent results  as  before. 

Now  suppose  another  distinct  case,  in  which  it  happens  that 
at  both  periods  w^e  purchase  100  A,  100  B,  and  100  C,  and  sup- 
pose the  price  of  A  rises  from  1.00  to  1.50,  and  the  price  of  B 
falls  from  1.50  to  1.2247  and  the  price  of  C  from  1.2247  to 
1.00.  Here  the  classes  [B]  and  [C]  together  make  a  fall  ex- 
actly the  rev^erse  of  the  rise  of  the  single  class  [A],  and  as  the 
individuals  in  each  of  these  classes  are  equal  in  number,  every 
rise  of  1  A  from  1.00  to  1.50  seems  to  be  met  by  a  fall  of  1  B 
and  1  C  from  1.50  to  1.00.  Therefore  we  should  expect  con- 
stancy. And  constancy  is  indicated  by  Scrope's  method,  which 
shows  that 

P,       150-H22.47  +  100  _ 

P,  ~  100  +  150  -f-  122.47  -  ^^^^■ 

But  if  we  apply  the  geometric  method  to  tliese  conditions  we 
must  give  these  weights  to  the  classes — to  [A]  v/l.OO  x  1.50 
=  1.2247,  to  [B]  v/1.50  X  1.2247  =  1.3554,  to  [G] 
\/ 1.2247  X   1.00  =  1.10r,7,  and  now 

P„ 


=     l/d-y--'"'  X  (x/2y-355-'  +  ii«'''  _  o.J)5»i):}02 , 

"- 1 

indicating  a  fall  by  a  trifle  less  than   0.0G9   per  cent.     To  get 
constancy  here  by  the   geometric   method  we  should   have  to 


DEVIATIOX    OF    THE   GEOMETRIC;    AVERA(;K  865 

weight  the  common  fall  of  [B]  and  [(']  as  2  to  the  rise  of  [A] 
as  1.  This  we  could  do  if  it  were  proj)er  to  weight  [B]  and 
[C]  by  doubling  the  geometric  mean  of  their  weights,  the  class 
[B]  l)eing  as  much  more  important  as  the  class  [C]  is  less  im- 
portant than  the  class  [A] .  But  we  can  form  no  general  prin- 
ciple of  weighting  of  this  sort.  For  instance,  in  the  preceding 
example  this  kind  of  weighting  would  still  weight  the  class  [B] 
as  two  to  [A]  as  1,  and  yet  with  this  weighting  the  geometric 
average  there  showed  about  the  same  error. 

Hence  it  appears  that  in  these  complex  cases,  even  on  the 
principle  of  the  geometric  method  itself,  Sc rope's  method  is  cor- 
rect, and  the  geometric  average  of  price  variations,  with  the 
best  weighting  we  know  of,  is  wrong  so  far  as  it  diverges. 

§  2.  In  the  examples  reviewed  the  ])reponderating  class  or 
classes  have  been  the  ones  that  fall  in  price,  and  the  geometric 
average  has  been  found  to  err  below  the  truth.  Making  [A] 
the  larger  class,  we  should  find  the  error  to  lie  on  the  other  side. 
These  facts,  added  to  what  we  already  know  about  the  deviation 
of  the  geometric  average,^  lead  to  the  inference  also  here  that 
when  the  prices  that  rise  above  the  general  average  are  those  of 
preponderating  classes,  the  geometric  average  of  price  variations 
yields  a  result  below  the  truth  ;  and  ivhen  the  prices  that  fall  below 
the  general  average  are  those  of  preponderating  classes,  it  yields  a 
result  above  the  truth. 

Luckily,  as  we  have  another  method  which  is  not  only  ex- 
actly correct  but  far  more  convenient,  we  are  not  so  much 
interested  in  the  error  of  the  geometric  method  as  we  were  in 
the  preceding  Chapter.  We  may  be  sure,  however,  here  as 
well  as  there,  that  with  moderate  price  variations  such  as  usually 
take  place,  the  geometric  average  will  not  much  deviate  from 
the  truth. 

§  3.  Some  extraordinary  cases  deserve  a  moment's  attention. 
Suppose  the  classes  [A]  and  [B]  are  equally  important  at  the 
first  period,  and  the  price  of  A  rises  from  1.00  to  1.99,  to  what 
figure  ought  the  })rice  of  B  to  fall  from  1.00  in  order  to  com- 
pensate for  that  rise  ? — always  supposing  that  the  mass  quanti- 

^  Cf.  other  instances  in  Notes  1  and  2  in  the  preceding  Section. 


360  I'Hi;    MKTHOD    FOR    CONSTANT    MASSES 

tics  are  constant.  We  can  now  answer  witliout  hesitation  :  it 
ought  t<)  fall  to  .01  ;  for  the  arithmetic  average  with  the  even 
weighting  of  the  first  })erio(l  indicates  constancy  under  these 
price  variations,  and  this  average  with  this  weighting  is  correct.^ 
Now  we  might  expect,  from  the  reasoning  in  Chapters  YII.  and 
YIII.,  that  according  to  the  geometric  method  this  fall  would  he 
made  out  to  be  too  great.  On  the  contrary,  the  couditions  are 
not  what  were  there  supposed,  and  they  require  the  geometric 
average,  applied  to  these  conditions,  to  indicate  a  rise — and,  be- 
cause of  the  enormous  variation  in  the  [)riee  of  B,  with  ccmse- 
quent  influence  upon  the  relative  sizes  of  the  classes,  a  consider- 
able rise — in  the  general  level  of  prices,  meaning  that  there  has 
not  been  sufficient  compensation.  In  fact  we  find  that  the  geo- 
metric average  here  indicates  a  general  rise  by  40.16  per  cent. 
And  if  we  suppose  that  B  falls  from  1.00  to  .01,  and  want  the 
compensatory  rise  for  A  so  as  to  make  money  constant  in  ex- 
change-value, instead  of  requiring  A  to  rise  from  1.00  to  1,99, 
the  geometric  method  requires  it  to  rise  from  1.00  only  to  1.463. 
The  truth  is  that,  when  dealing  with  such  enormous  price  vari- 
ations of  the  smaller  class,  as  shown  in  similar  instances  in  the 
preceding  Chapter,  the  geometric  method  becomes  wholly,  un- 
workable. To  shc)w.  this  we  may  comi)arc  the  results  given  by 
the  geometric  average  with  its  proper  weighting  with  the  true 
results,  indicated  by  the  arithmetic  average  with  even  weighting, 
when,  the  classes  [A]  and  [B]  being  equally  large  at  the  first 
l)eriod,  and  the  mass-quantities  remaining  constant,  the  price  of 
A  is  sup]M)sed   in  all  cases  to  rise  from  1.00  to   1.99,  and  the 

2  As  is  reiidcnnl  plain  by  tiie  following  sfhenia  : 

I    100  A  (c$  1.00    7.092  B"  @  14.10    I  100  for  [A]     100  for  [B]  —200, 
II   100  A  (<$  1.99    7.092  B"  @      .141  |  199  for  [A]         1  for  [B]  —200, 
in  which  the  mass-units,  A  and  B",  are  equivalent  over  both  the  periods  together, 
and  the  same  total  sum  is  paid  at  each  period  for  purchasing  these  equivalent 
mass-units.     (To  have  even  weighting  over  both  the  periods  together,  the  condi- 
tions would  have  to  be : 

I   100  A  @  1.00  '  100  B  @  14.10    I  100  for  [A]     1410       for  [B]  —  1510, 
II   100  A  @  1.99     100  B  @      .141  |  199  for  [A]         14.10  for  [B]  —  213.10. 

Here  the  geometric  average  with  even  weighting,  »  1.99  X  .01  =0.1411,  indicates 

a  fall  of  85.89  percent.;  but  also  the  arithmetic  average  with  the  weighting  of 

21'?  1 
the  first  period  indicates  the  same  fall,  for  "■_  '    -- 0.1411.) 


I)p:viati()N  of  tmk  (jeometkic  average  3()7 

price  of  B  is  sup])()S('(l  to  fall  variously  to  the  following-  figures, 
stated  in  the  Hrst  eoluniu,  the  true  average  being  stated  in  the 
next  column,  and  the  geometric  average  in  the  last : 

1.00  1.495  1.4958 

.81  1.400  1.4021 

.64  1.315  1.3200 

.49  1.240  1.2502 

.36  1.175  1.1947 

.25  1.120  1.1563 

.16  1.075  1.1403 

.12  1.055  1.1436 

.09  1.040  1.1562 

.04  1.015  1.2250 

.01  1.000  1.4016 

.005  0.9975  1.4954 

Here  the  geometric  method  remains  approximately  correct  till 
^.^  descends  beyond  .50  and  its  variation  becomes  the  greater 
of  the  two,  while  [B]  becomes  less  than  half  as  large  as  [A] 
over  both  the  periods,  after  which  it  departs  appreciably  from 
the  truth.  A  strange  thing  is  that  after  fi.^  reaches  about  .15, 
the  further  it  descends,  the  more  the  geometric  average  rises. 

An  objection  previously  urged  against  the  arithmetic  average 
was  that  with  even  weighting  it  permits  of  no  compensatory  fall 
of  B  after  A  has  risen  to  2.00.  That  was  on  the  supposition  of 
even  weighting  over  both  the  periods  together.  With  condi- 
tions permitting  of  even  weighting  over  both  the  periods  to- 
gether there  should  be  possible  a  compensatory  fall  of  B  for 
every  rise  of  A.  But  if  the  weighting  is  even  only  at  the  first 
period,  and  if  the  sizes  of  the  classes  rise  and  fall  with  the  rise 
and  fall  of  their  prices,  there  being  no  change  in  our  purchases 
of  them,  there  is  no  reason  why  beyond  a  certain  point  in  the  rise 
of  A  there  should  be  no  compensatory  fall  of  B  possible.  That 
objection  against  the  geometric  average  was  accompanied  by  an 
argument  for  the  geometric  average.  We  now  see  that  under 
the  circumstances  supposed  the  geometric  average  is  in  this  re- 
spect no  better  than  the  arithmetic,  each  with  its  proper  weight- 
ing. It  is  even  worse,  as  it  takes  away  the  possibility  of  com- 
pensation at  an  earlier  point  in  the  rise  of  A.  Moreover  it  is 
peculiar  in  that  after  A  has  passed   the  point  where  the  possi- 


:M]X  rilK    METHOD    FOR    COT«^STANT    MASSES 

bilitv  ofconipletL'  CDnipensatiou  ends,  the  descent  of  B  gains  its 
maximum  compensation  before  reaching  zero. 


§  1.   In  closing  this  Chapter   some  tests   may  be  employed 

similar  to  those  at  the  end  of  the  preceding. 

I  100  A  @  1.00     100  B  @  1.00  1  100  for  [A]     100  for  [B]  —  200, 

II  100  A  @  1.50     100  B  @    .60  I  150  for  [A]       60  for  [B]  —  210, 

III  100  A  @.  1.00     100  B  @  1.00  I  100  for  [A]     100  for  [B]  —  200. 

Here  it  is  evident  that  the  exchange-value  of  money  is  the  same 
at  the  third  period  as  at  the  first,  whatever  it  be  at  the  inter- 
mediate stage ;  and  this  constancy  at  the  third  period  compared 
witli  the  first  should  be  indicated  by  the  re.sults  of  the  two  vari- 
ations. The  weighting  at  the  first  and  at  the  third  period  is 
even,  but  at  the  second  period  it  is  5  for  [A]  and  2  for  [B] . 
Now  if  we  use  the  arithmetic  average  in  both  measurements, 
and  each  time  with  the  weighting  of  the  earlier  or  first  period 
of  the  two  eom])ared,  that  is,  in  the  first  price  variations  with  the 
even  Aveighting  of  the  first  period,  we  get  an  indication  of  a  ri.se 
of  5  per  cent,  (i'f +  f!  =  ftr  =  ^-^^'"0  5  ^"'*^  "^  *^^®  second 
price  variations  with  the  uneven  weighting  of  the  second  period 
we  get  an  indication  of  the  inverse  fall  of  4,77  per  cent. 
(i|5  X  I  -h  2  X  f ;  =  If  =  0.9523)  ;  and  these  two  together 
indicate  the  .same  level  of  prices  at  the  third  period  as  at  the 
first.  And  exactly  the  same  indications  are  given  by  the  har- 
monic average  of  the  first  price  variations  with  the  weighting  of 

/             7  2 1  \ 

the  second  period  (   ~ TXo 5  =  oo  )  '  '^"^^  ^^y  ^^'^'  harmonic 

av^eragc  of  the  second  ])rice  variations  with  tlie  even  weighting  of 

/      2  2()\ 

the  third  period  (  ^        ^  =  ^^     J — as   also   directly   by   Scrope's 

method  of  com))aring  the  total  sums.'  And,  once  more,  the  correct 

>  The  same  re.sults  could  also  be  obtained  by  combining  these  averages  differ- 
ently and  using  the  same  weighting  throughout: — thus  either  by  using  even 
weighting  (of  the  earlier  and  of  the  later  period)  witii  the  aritiimetic  average  of 
the  first  variations  and  the  harmonic  average  of  the  second  ;  or  by  using  the 
weighting  5  for  [A]  and  2  for  [H]  (that  of  the  later  and  of  the  earlier  period) 
with  the  harmonic  average  of  the  first  variations  and  the  arithmetic  average  of 
the  second.     Rut  these  an"  methods  above  rejected. 


TKST   (ASKS  ;i(59 


final  result  is  obtained  also  by  iisinir  the  geometric  average,  in 
both  cases  with  the  same  Weighting,  M'hieh  is  l.oHH  for  [A] 
and  1  for  [B] .      With  this  weighting  the  geometric  average  of 


2.5811/ 


the  first  price  variations  is  |/  (|y-^*"  X  f  =  1.051G,  indicat- 
ing a  rise  of  5.16  per  cent.,  and  of  the  second,  "■''|^^(2y-5''ii  y^  5 
=  0.9508,  iudicatmg  a  fall  of  4,92  per  cent.  It  would  seem  as 
if  the  same  weighting  ought  to  be  used  in  both  the  averagings, 
since  in  each  set  of  variations  the  only  difference  in  the  weight- 
ings is  the  order  of  their  occurrence.  Still  the  geometric;  aver- 
age does  not  give  such  true  indication  in  each  set  of  price  vari- 
ations as  do  the  other  two  methods.^ 

We  perceive,  however,  that  the  error  of  the  geometric  method 
above  the  truth  in  the  first  price  variations,  where  the  rising- 
price  is  of  the  predominating  class,  is  exactly  counterbalanced 
by  the  error  of  the  geometric  method  below  the  truth  in  the 
second  price  variations,  where  the  variations  are  reversed  and 
the  falling  price  is  of  the  predominating  class  ;  for  1.0516  :  1.05 
::  0.952.3  :  0.9508. 

If  the  weights  were  uneven  at  the  first  and  third  periods,  or 
altogether  so  different  as  to  make  the  classes  very  unequal  in 
size  over  both  the  periods  together,  in  each  comparison,  while 
the  price  of  the  smaller  class  varies  consideral)ly,  the  error  of 
the  geometric  method  above  or  below  the  truth  in  each  measure- 
ment would  be  greater  than  in  the  case  here  cited,  and  might  be 
considerable.  Yet  the  same  counterbalancing  would  always 
take  place,  and  the  indication  for  the  third  period  thi-ough  the 
intermediate  one  would  still  indicate  constancv. 

§  2.  With  two  or  more  intermediate  periods  the  geometric 
average  does  not  necessarily  give  the  right  indication  for  the 
last  period.     For  example  : 


I  100  A  @,  1.00  100  B@  1.00 

II  100  A  @  1.50  100  B@    .75 

III  100  A  @    .661  100  B@  2.00 

IV  100  A  (a).  1.00  100  B  (a),  1.00 


100    for  [A]  lOU  for  [B]— 200, 
150    for  [A]       75  for  [B]— 225, 

66|  for  [A]  200  for  [B]— 266|, 

100    for  [A]  100  for  [B]— 200.  . 

2  The  true  final  result  is  also  obtained  through  intermediate  errors  in  two  other 
ways — namely,  by  averaging  both  sets  of  variations  arithmetically  with  the 
weightings  of  the  later  periods,  or  harmonically  with  tlie  weightings  of  the 
earlier  periods.  But  these  are  methods  also  above  rejected.  Thus  there  are  seven 
ditferent  ways  of  ol)taining  the  known  final  result. 
24 


."JTd  riiK  MK'riioD   i"(»i;  coNSTAN'r  sims 

Here  tlu' u'cuniL'ti'ic  iik'IIkhI  <i-ives  tlu'  folluwino;  indications  : 

1st  vnrinticn  'X'' i^'''''  x|  =  1-1256, 
•id  vaviati.H.  ■""■"i/'^|~X  (1)'--*'  =  1.1917, 
:5d  v:.ri:.ri(.n    'fXfx  (^f'^  =  0.7475. 

These  form  tlic  series  of"  inde.\-nunil>ers  100,  112. 5fi,  134.14, 
100.27  ;  and  so  this  method  is  wrong  at  the  end  by  an  error 
0.27  per  cent,  above  tlie  trnth.  As  indicated  in  the  last  column, 
Scrope'.s  nietliod  uives  the  index-numbers  100,  112.50,  133.33, 
100;  wliich  show  the  coi-icct  variations  to  be:  1st  1.125,  2d 
1.1852,  3d  0.75. 

In  Scrope's  method  is  perceived  the  merit,  as  there  has  al- 
ready been  o(!casion  to  remark,  tliat  it  universally  stands  Pro- 
fessor Westergaard's  general  test. '  It  is  always  consistent  with 
itself  in  direct  and  in  in<lirect  comparisons  over  any  number  of 
periods.  It  stands  one  and  all  of  the  tests  to  which  we  can  sub- 
je(^t  it.  (  ontined  to  the  cases  wlien  the  mass-quantities  are  con- 
stant, tliis  method  is  absolutely  pei'fect.  And,  consequently,  so 
also  the  two  methods  of  averaging  the  [)rree  variations  into  which 
it  can  be  exactly  analyzed. 

Thenfore,  possessed  of  this  pei'fect  yet  simple  and  most  con- 
venient method,  we  have  no  interest  here  in  })ursuing  fiu'ther 
the  investigation  of  the  slightly  deviating  error  in  the  geometric 
method.  We  may,  however, — to  extend  a  remark  already  made 
— be  sure,  here  as  well  as  in  the  preceding  Chapter,  on  account 
of  the  many  counterbalaucring  iniiuences,  the  geometric  method 
is  not  likely  in  a  long  series  to  be  f-ir  wrong  at  any  period. 

S  ■<■  ('li;i|.I.  W  Scrt.    \l.   i^  7. 


CHAPTER  XII. 

THE  UNIVERvSAL  MKTIK  »I). 

T. 

§  1.  The  conditions  presupj^o.sed  by  h<»tli  the  prcccdinir  ;u"_i>u- 
meuts  are  unlikely  to  be  met  Mdth  in  ])raotice.  The  tirst  argu- 
ment, in  supposing  the  sums  of  money  to  he  constant  in  sjnte  of 
the  price  variations,  can  have  application  only  if  prices  vary 
through  changes  in  sup]ily.  The  second,  in  sup])osing  the  mass- 
quantities  to  be  constant  in  s]iite  of  the  })rice  variations,  can 
have  application  only  if  prices  vary  through  changes  in  de- 
mand. Neither  of  these  changes  is  likely  to  take  place  alone. 
Prices  generally  vary  througii  changes  both  in  supply  and  in 
demand.  Both  these  conditions  must  be  admitted  ;  and  only  that 
argument  is  conijjlete  which  takes  them  V)oth  into  consideration. 

Or  the  user  of  the  iirst  argument  can  defend  his  position  (»nly 
by  claiming  that  the  weighting  should  be  according  to  the  smaller 
money-values  (or,  more  properly,  the  smaller  exchange-values) 
at  either  period.  And  the  user  of  the  second  argument  can  de- 
fend his  j)osition  only  by  claiming  that  the  weighting  should  de- 
pend upon  the  smaller  mass-quantities  at  either  [)erio(l.  Each 
must  rely  upon  something  that  is  common  to  both  the  periods, 
eliminating  all  the  rest  of  the  same  kind. 

The  practical  and  theoretical  obje(^tions  to  these  ])ositions  have 
already  been  set  forth  in  Cha})ter  IV.  Section  \.  Their  im- 
propriety is  especially  apparent  wlien  we  |)lace  them  side  l)v 
side.  For  which  of  them  has  more  reason  for  it  than  tlie  other? 
If  we  want  the  same  material  world  at  l)oth  periods,  do  we  not 
equally  much  want  the  same  economic  world  at  both  periods? 
We  can  not  have  both  of  these  things  together.  And  trial 
shows  that  on  posited  variations  of  prices  and  of  mass-quniitities 

:'.71 


372  TllK    INIVERSAl,    MHTlloI) 

the  two  methods  give  very  diverfj;ent  results — sometimes  the 
former  giving  the  higher  result,  and  sometimes  the  latter,  and 
more  or  less  so,  and  sometimes  botli  giving  nearly  the  same  re- 
sults, without  order  or  principle.  And  to  use  both,  (h'awing 
some  mean  between  their  results,  M'ould  convey  un  sj)eeial  mean- 
ing, and  has  no  justification. 

Or  again,  the  user  of  the  first  argument  might  defend  his  posi- 
tion by  drawing  a  mean  between  the  total  money-values  of  a 
class  at  each  period,  and  treating  this  mean  total  money-vahie 
as  if  it  were  the  total  money-value  at  each  period, — and  d<»iiig 
so  with  every  class,  he  would  weight  tliem  accordingly.  And 
the  user  of  the  second  argument  might  defend  his  position  like- 
wise by  dra^^'i^lg  a  mean  between  the  total  mass-quantities  of  a 
class  at  each  period,  and  treating  this  mean  total  mass-quantity  «,s 
if  it  were  the  mass-(|uantity  at  each  period, — and  doing  so  with 
every  class,  he  would  employ  these  niass-(juantities  as  the  basis 
of  his  weighting  (or  would  simply  a})ply  to  them  Scrope's 
method). 

Now  if  the  user  of  the  first  argument  employed  the  geometric 
mean  between  the  total  money-values  of  every  class  at  each  of 
two  periods  as  the  weight  of  every  class,  he  would  be  doing 
merely  what  has  already  been  rec(mimended  in  Chapter  IV. 
He  would  do  well,  however,  to  avoid  recommending  his  ])ro- 
cedure  on  the  ground  of  an  rw  //'.  The  total  money- values  of 
the  classes  are  different  at  each  period,  and  wliat  we  want  is  not 
what  might  have  been  a  similar  state  of  things  if  a  mean  total 
money-value  had  existed  in  every  class  at  both  j)eriods  ;  but,  as 
explained  in  the  earlier  Chapter,  what  we  want  is  the  ninnl>er 
of  individuals  in  every  class  that,  given  the  facts  as  they  are, 
have  the  same  exchange-value  over  both  the  j)eriods.  And  if 
the  user  of  the  second  argument  employed  a  mean  between  the 
total  mass-<piantities  at  each  jx'i'iod,  he  would  be  einj)loylng  a 
method  also  admitted  in  that  Chapter  as  a  tenable  one.  But  he, 
too,  ought  to  find  some  other  reason  for  iiis  position  than  an  «.s 
if.  As  for  the  kind  of  mean  to  use  here,  everything  we  have 
learned  points  to  the  geometric  mean  as  the  proper  one.  But 
we  shall  later  find  that  the  arithmetic  mean  irives  almost  similar 


TWO    MpyPHODS    SUGGESTED  .'>73 

results,  wlicrctore  its  greater  convenience  is  a  recommendation 
for  its  practical  employment. 

§  2.  Thus  all  the  methods  as  yet  in  these  pages  examined  that 
have  any  claim  to  consideration  as  theoretically  reasonable  and 
as  likely  to  yield  truthful,  or  nearly  true,  answers,  reduce  to 
these  two  :  (1)  the  geometric  averaging  of  the  price  variations 
with  weighting  according  to  the  geometric  mean  of  the  full  total 
money-values  at  both  periods — which,  as  before,  for  brevity,  we 
shall  call  simply  the  geometric  method  ;  and  (2)  Scrope's  method 
applied  to  the  geometric  means  of  the  full  mass-quantities  at  both 
periods — ^vhic'h,  again  for  brevity,  we  shall  call  Scrope's  emended 
method. 

For  the  first  of  these  the  recommendation  is  that  it  takes  into 
consideration  the  conditions  at  both  periods,  correctly  weights 
the  classes  and  uses  the  best  average  we  have  for  averaging  the 
price  variations  of  the  economic  individuals,  whose  relative  num- 
bers have  been  determined  in  the  weighting.  The  objection  to 
it  is  that  the  geometric  average  ceases  to  possess  the  virtue  of 
the  geometric  mean  when  it  is  applied  to  more  than  two  classes, 
or  to  two  unequal  classes.  It  is  true  that  all  commodity-classes 
fall  into  two  general  classes :  those  whose  prices  rise  above  the 
average  variation,  and  those  Avhose  prices  fall  below  it, — not  to 
mention  a  third  class,  whose  prices  vary  with  the  average,  the 
presence  of  ^vhich  class  is  indiiferent.  And  in  practice  those 
two  general  classes  may  mostly  be  nearly  equal  in  size.  Hence 
in  jiractice  the  geometric  average  is  not  likely  to  depart  much 
from  the  truth.  Still,  we  have  seen  that  when  the  classes  are 
very  unequal  and  the  price  variations  are  very  great,  this  aver- 
age may  deflect  considerably.  Therefore  we  should  prefer  not 
to  have  to  rely  on  this  method  alone. 

The  second  of  these  methods  has  in  its  favor  that  it  likewise 
takes  into  consideration  the  conditions  at  both  periods,  and  that 
it  avoids  the  use  of  the  geometric  average,  using  only  the  geo- 
metric mean,  so  that  it  escapes  perversion  arising  from  that 
source  in  the  extraordinary  cases  of  great  inequality  in  the  sizes 
and  of  great  variation  in  the  })rices.  The  principal  argument 
for  it  is  that  it  has  been  found  to  be  the  fundamental  method 


374  riii:   inivkksai,  method 

imdci-lviiiii-  Ixitli  the  |);ii-ti:il  nu'tliiids.  For,  being  the  right 
iiicthdd  hdth  for  constant  nioney-sinns,  wiien  tiie  mass-quanti- 
ties vary,  and  for  constant  nuiss-cinantitif^"^,  when  the  money- 
sums  vary,  why  should  it  not  be  extendible  to  all  cases,  and  be 
the  true  universal  method?  Unfortunately  only  one  of  those 
|)artial  methods  was  found  to  be  perfect.  And  the  very  one  by 
which  this  niethod  is  suggested  was  found  not  to  be  perfect. 
Hence  there  is  a  proi)ability  of  this  method  also  not  being  perfect 
in  "all  cases. 

There  is,  however,  still  another  Avay  <»f  deducing  a  universal 
method  from  the  two  partial  methods, — adding  a  third  method 
with  claim  upon  oin-  attention  as  likely  to  give  approximation 
to  the  truth.  This  new  method,  by  slightly  altering  each  of 
those  methods  in  the  same  manner,  apjilies  them  to  all  ])()ssible 
cases,  getting  tlie  same  result  always,  whether  it  modifies  the 
one  or  the  other.  And  this  alteration,  or  modification,  in  each 
case,  .respects  the  principle  of  simple  mensuration,  that  we  must 
at  both  ])erio(ls  l)e  dealing  with  the  same  whole,  or  with  similar 
wholes,  allowing  the  details  to  be  different.  Hence  this  method 
seems  t(»  lia\c  iiiticli  in  its  favor.  Since  it  modifies  the  imper- 
fect one,  there  is  some  hope  that  it  escapes  the  imperfection. 
Whether  it  does  so,  will  be  seen  in  the  sequel.  We  must  now 
develop  this  method.  Then  we  may  test  it,  and  conn)are  it 
w  ith  the  othci'  methods. 

II. 

1^  1.  Kotli  the  partial  methods  examined  in  the  preceding 
('liapters  start  out  with  the  sanu^  injimetion,  that  we  shotdd  ///*'/ 
ill  (ill  flic  c/d.s.-tc.s  iiKi.s.s-iiiufx  (hat  lid  re  the  sdinv  <:vchaiif/<-r(iliir 
orer  hofli  flic  jicriods  fot/cflirr.  This  having  been  done,  the  first 
nietho<l,  a|)|)lied  to  conditions  where  the  sums  of  money  spent 
on  the  classes  are  not  the  same  at  both  periods,  becomes  this  : 
iiicdsiirc  the  consfanei/  or  variation  of  the  purchasing  power  of 
monoji  by  the  consfanci/  or  iviriation  in  the  total  number  of  theae 
nuiHs-unHs  jji(rcli((s('il  or  purdiasahle  at  each  period  with,  a  (jireii 
total  nam  of  uioueij  spent  in  the  same  proportions  a^  the  total  sums 
adualtji  were  spent  at  each,  period.     Of  course  we  may  take  one 


A  Til  I  HI)  MK'inoD  .><r> 

of  the  total  sums  as  it  actually  was,  and  reduce  the  other  to  it. 
The  second  method,  a})j)lied  to  eoiiditions  where  the  mass- 
quantities  purchased  of  the  classes  are  not  tlie  same  at  hoth 
periods,  becomes  this  :  mea.mre  tJir  ronKtaiwy  or  var'mfion  of  f/ir 
exchange-value  of  money  inversely  by  the  constancy  or  tmriatiou 
in  the  total  sum  of  money  needed  at  each  period  to  purchase  a  given 
total  number  of  these  mass-unit'^  i)i  the  .^anie  proportions  as  fhr 
total  numbers  of  them  actually  ivere purchai<ed  at  each  period.  Of 
course,  again,  we  may  take  the  one  of  the  total  numbers  as  it 
actually  was,  and  reduce  the  other  to  it.  The  measurement  of 
the  constancy  or  variation  of  prices  in  general  is  the  inverse  of 
these,  that  is,  by  inversion  in  the  first  case  and  without  inver- 
sion in  the  second.  Both  these  methods,  to  repeat,  applied  to 
tlie  same  conditions  where  both  the  sums  of  money  and  the  mass- 
(piantities  vary  from  period  to  period,  yield   the  same  result. 

These  two  methods,  which  really  form  one  l)ipartite  method, 
call  for  illustration  first  by  numerical  exam])k's.  Suppose  we 
find  this  state  of  things  : 

I  90  A  @,  1.00       70  B  Oi)  1.00  I     90  for  [A]     70  for  [B], 
II  80  A  @  1.50     150  B  (o     .50  |  120  for  [A]     75  for  [B], 

This  admits  of  conversion  into  tiie  followino- : 


1 90  A  0  1.00    40.415  B'^ @  1.732  —  130.415 
II 80  A  (7»  1 . 50    86. 605  B''  0'\    . 866  —  166. 605 


90  for  [A]     70  for  [B]— 160, 
1 20  for  [A]     75  for  [P.]  —  195, 


in  which  the  mass-units,  A  and  B",  have  the  same  money-valiu', 
and  the  same  exchange-value,  over  both  the  periods  together. 
Now  at  the  first  period  we  bought  1 30.41 5  such  mass-units  for  1 60 
money-units,  and  at  the  second  we  bought  166.605  such  mass- 
units  for  195  money-units.  Therefore  at  the  second  period  we 
could  buy,  spending  our  money  in  the  same  proportions  as  we 
did  then  actually  spend  it,  ]o().70l  such  mass-units  for  160 
money-units  (as  we  learn  by  the  simple  use  of  the  rule-of-three). 
Thus  the  purchasing  power  of  160  money-units  has  risen  fnmi 
purchasing  at  the  first  ])eriod  l.">0.415  to  purchasing  at  the 
second  ])eriod  lo6.701  mass-units  that  are  ecpiivalent  over  both 

the  ])eriods.      Hence  its  variation  has  lu'cn         '         =1.0482; 
'  •  1:30.415 


37()  I'm-;   imnimisai,   MiynioD 

and   the  variation    <if"  prices  has    hccn  '  =  ()A)i')M),  iiuli- 

eating-  a  tall  of  prices  bv  4.()<)  i)er  cent.  And,  again,  at  the 
first  period  we  gave  KK)  money-units  for  180.415  such  mass- 
units,  and  at  the  second  we  gave  195  money-units  for  166.605 
such  ni;i~;s-units.  Therefore  at  the  secDnd  period  \vc  gave  152.64 
money-units  for  130.415  sucii  mass-units,  in  the  same  {propor- 
tions as  we  actually  did  then  ])urcliase  them.  Tluis  this  mass- 
{|uantityof  l.i0.415  mass-units  eijuivalcnt  over  both  periods  has 
fallen    in    pi'ice  from  IdO   at  the  first   period  to    152.64  at  the 

1          1     !  •  .•  /-  .1  I'll-    152.64 

second,  and   tlie   variation  of  tlie  u'eneral  price-level   is 

^  '  160 

=  0.1>54(),  indicating  a  fall   by  4.60  per  cent.,  the  same  as  be- 
fore.     It  may  be  added  that  on  these  data  the  geometric  method 
indicates   a   fall   of   prices    by  4.49    per   cent.;'    and   Scrope's 
emended  method  indicates  a  fall  by  4.71  per  cent.^ 
Another  example  may  be  supposed  as  follows  : 


I  75  A  @,  1.00       70.71  B  @  1.00 
II  (50  A  (W,  1.50     13,3.^      B(a}    .75 

This  mav  be  converted  into 


75  for  [A]       70.71  for  [I^], 
90  for  [A]     100       for  [B]. 


I  75  A@,1.00  50      IVU^  1.4142—125       |  75  for  [A]    70.71  for  [B]— 145.71, 
II60A@1.50  94.28irr'^  1.0(300—154.28  <.IOfor[A]  100       for  [B]— 190.00, 

in  which  the  mass-units,  A  and  B",  are  e([iiivalent  ov(!r  both 
the  ])eri()ds.  At  the  first  period  we  got  125  such  mass-units  for 
145.71  money-units,  and  at  tlu;  second  154.2<S  of  them  for 
190.00.  Therefore,  in  the  same  proportions,  at  the  second 
period  we  should  get  118.31  such  mass-units  for  145.71  money- 
units;  and  so  the  j)urcliasing  power  of  this  sum  of  money  has 
fallen  from  purchasing  125  to  purchasing  1  1<S..']1  ;  and  inv(!rsely 

125 
the  general    level   of   ])rices  has  varied  thus:        ,  .      =  1.0565, 

which  indicates  a  rise  by  5.65  per  ceiil.  Again,  at  the  first 
period  we  paid  145.71  money-units  for  125  such  mass-units, 
and  at  the  second  190.00  for  154.2S.      Therefore  at  the  second 

'  Tliis  is  s]ifi;litly  !il)()vu  tlie  utlier  rt'sult.     Tlu-  wcifilits  arc  1.4:U2  for  [A]  and 
1  for  [!>].     Notice  tliat  the  rising  price  is  of  the  j)repon(lerating  chiss. 

-'  On  the  arithmetic  means  Scrope's  method  indicates  a  fall  by  (i.41  per  cent. 


A    riiii;i>   MirnioK  .U  i 


period  we  had  to  pay,  in  the  same  proportions  in  wiiich  we 
actually  did  spend  our  money,  l."),3.<)4  money-units  for  125  mass- 
units.  Thus  the  price  of  125  such  mass-units,  which  an;  ecpiiv- 
alent  over  both  the  periods,  has  risen  from   145.71  to  153.87, 

153,94 
and  the  price  variation  i'^^  ir'71  =l.<^5n5,   likewise    indieatiug 

a  rise  by  5.05  per  cent.  Here,  we  may  again  add,  the  geometric 
method  indicates  a  rise  by  5.64  per  cent.;''  and  Scropc's  method 
also  indicates  a  rise  by  5.64  per  cent.^ 

§  2.  That  these  two  ways  of  making  the  calculation  univer- 
sally agree,  and  coalesce  into  one  method,  may  be  demonstrated  as 
follows.  p]mploying  our  usual  symbols,  we  construct  this  com- 
prehensive schema,  representative  of  any  possible  state  of  things  : 

I  ri  A  @  «!     J/i  B  @  f?!     aiC@}',  —  •''i  +  i/i  +  ;i  + 

j:,rt,  for  [A]     ?/,/?!  for  [B]     2,},  for  [C] —  xyh  "*■  ^i^i  +  hYi  + > 

x.,a^  for  [A]     y,[i.,  for  [B]     2,7,  for  [C] —  x./i.,  -\-  y^S,  +  2272  -f- , 

in  which    the   mass-units.   A,  B,  (',  ,  of  the  classes,    [A], 

[B],  [C] ,  are  the  customary  commercial  ones,  and  x,  y, 

z,  ,  represent  the  numbers  of  them  that  are  bought  and  sold 

at  the  periods  indicated  by  the  numerals  attached  to  them,  and 

n.,  /5,  )',  ,  represent   the   prices   of  these  mass-units   at   the 

periods  similarly  indicated.  The  number  of  classes  may  be  ex- 
tended indefinitely,  but  must  be  the  same  at  both  periods.  As 
the  mass-units  are  various,  we  must  reduce  them  to  equivalence 
over  both  the  periods.  We  may  take  the  mass-unit  of  any  one 
of  them,  say  of  [A] ,  as  our  unit,  and  reduce  the  rest  to  equiva- 
lence with  it.  Represent  the  mass-unit  of  [B]  that  is  equiva- 
lent over  both  periods  to  A  by  B/'  and  its  prices  at  each  of  the 
periods  by  ,?/'  and  fij'  respectively  ;  and  treat  the  equivalent 
mass-unit  of  [C]  in  the  same  way.  The  first  half  of  the  above 
schema  then  becomes  the  following  : 

3  Almost  exactly  like  the  other,  but,  this  time,  slightly  below.  Notice  that 
the  weighting  is  almost  even,  but  slightly  preponderating  on  the  side  of  the  fall- 
ing price,  the  weight  for  [A]  being  82. Ki  and  tliat  for  [B]  84.09. 

*  These,  however,  differ  in  further  decimals.  For  their  closeness  the  near 
evenness  of  the  weighting  will  also  be  found  to  be  the  reason. — On  the  arithmetic 
means  Scrope's  method  indicates  a  rise  by  4.8()  per  cent. 


37S  thp:  rxiVEHsAi.  .Mi/nion. 


I  .T.  A  @  a.    5v  ^''  fe'  ^/'      S  <-"'  <!i>  y" 
P\  /I 

II  X,  A®n,   0  B^  @  r},^      J  J  C^  r«^  )'/ 


Here  the  conditions  are  that  ,i/','i.,"  =  /'/'/V  =  =  "Z^-  r  ^'"^^ 

.'2  /'2       /2  / 


also  tlmt ',^,,  = 't^  ,    ",,  =    ',  and  so  on.      From  tlu' first  coiuli- 

/^i         <^i    /'i         T\ 

9  "      «  « 
tion   we  derive  '7^,,=  ^ri)  wherefore,  bv  means  of  the  second 

coiHUtion,   ,-=  yy^:,,  whence  ,:^/=      -     '.      In  a  snnilar   man- 
t'^x       Pi   '  Pz 

ner  we  obtain  ^9/'-=  '^^ .     Thns  we  have  ;?/'=  J"*"'^i ,  and 

Pi  ^        /'2 

jV'=  j"^;  wherefore 

^     Pi  _  

yj^i  _  vA^  _  ,.    I/V2 


and 


;^  "  / 5  "2  ^  • 

P2  ^  <'-x'l<2ii-i  ^«iS 

And  treating  the  prices  of  [C]  in  the  same  Avay,  we  obtain 
and 

V2  _ .,    I  ro'-i . 
r2"~"">^«2' 

and  so  on   with  all   the  other  classes.      By  snbstitnting-  these 
valnes  in  the  last  sciiema,  it  becomes  : 

1   r,  A  (a];  a,     y,\  '^"''^  \V'  @  /?/''     z,  V^  '^''  C'^  @  )'/'    

•'■.'+  y\  \ h  ^2 A        -r  , 


A    I'liiiM)   Mi;rm»i)  oTJ) 

the  second  Imlfof  tlie  sclu'iiia  rciuaiiiiiig  the  same  as  in  the  first. 
Here   A,  B",  C",  arc  mass-units  equivalent  over  both  the 

^iJ'-~+ s"(*J»  mass-units  for  •'•,«! +i/w^i  +  ^r/'i  + money- 

units;    and    at     the    second     j)criod     we     get    -t-'^+ya  ^ h 

J'''*'^-!- such  mass-units  for  x,a„-\-i/.,f-i.,-^z.,y.,+  raoney- 


units.  Therefore,  in  the  same  propin'tious,  we  should  get  at  the 
second  period  (as  we  learn  by  the  rule-of-three) 

«2S+  !f2i^2+  V/+    

sucli  mass-units  foray/j  -f-  lf^f:!^  +  z^Y^  +  money-units.    Thus 

the  purchasing  power  of  this  sum  of  money  has  varied  from 
purchasing  the  former  number  to  purchasing  the  latter  number 
of  mass-units  equivalent  over  both  the  periods ;  and  the  pur- 
chasing power  of  all  sums  of  money,  or  of  money  in  general, 
that  is,  the  exchange-value  of  money  in  all  the  other  things,  has 
varied  in  the  same  proportion.  And  the  general  level  of  prices, 
varying  inversely,  has  varied  as  is  thus  represented, 

(-^+^^A/^;^:+Wg:+ )(^.-.+.A+^.n+ ) 

(■•■■' + '"V!S+  ^'^S + )  ("■"■ + '■"'' + '•'■  "^ ) 

Again,  at  the  first  period  we  give  .<//, -f  i/^,ii  -|- ^,/'|  +  money- 


Pi 


380  THK   rM\i:i;sAi,   Mini  ion 

units  for  ./•,  +  '/,  *,r  ^'  "  +  r,  .      '  "  +  such   mass-units,  and 

at  the  second  period  we  give  ■»■.//■.,+ //a^, +  -../'•  +  money- 
units    for    .v.,  -j-  II ,  J' ^—  4- z ,  J      '^  -{-  sucli     mass-units. 

Therefore,  in  the  same  pro})ortions,  we  sliould  at  the  second  period 
have  to  <^ive 

money-units  for  :?•,«,  -|-  ?/,  A'  "  '  -f-  .:,  .    '^''"  +  such    mass- 

units.  Thus  the  price  of  this  luuuber  of  mass-units  equivalent 
over  both  periods  has  varied  from  the  first  to  tlie  second  sum, 
and  in  the  same  proportion  lias  varied  the  price  of  any  given 
number  of  such  mass-units  })urchased  at  each  period  in  the  same 
proportions  as  each  of  the  totals  at  those  periods  actually  were 
purchased.     Therefore 

p, ^  (■''  ^-''^  V'^  +  Wg;  + )  ('y.  +?»'.  +  %'^  + ) 

*"■    (-.+».a/^:  +  W;*  + )  (»■,«.+ ..ft +  w,  + )' 

which  is  the  same  as  the  preceding,     (i.  E.  D. 

Above,  in  Chapter  IX.  (Section  I.)  we  criticized  the  argu- 
ment from  compensation  by  equal  mass-quantities  and  the  argu- 
ment from  compensation  by  equal  sums  of  money  because, 
although  modelled  on  correct  methods  of  measuring  variations  in 
a  particular  exchange-value  of  money,  which  give  the  same  result 
from  two  (>j)posite  points  of  view,  those  arguments,  as  hithert(^ 
employed,  gave  different  results,  disclosiug  error  somewhere. 
We  then  saw  that  the  error  lay  in  the  method  usually  adopted 
of  employing  the  argument  from  compensation  ])y  c((ual  mass- 
quantities.  Thereupon  we  determined  the  right  method  of  em- 
ploying the  argiunent  from   conqx'usation   by  e(|ual    mass-quan- 


A    iim;!)   Mi;rii(iii  :!S1 

tities,  Wc  now  find  tluit  tlii^  riuht  use  (»(' tliis  niymnciit  and  tlic 
right  use  of  tlu'  other  ai'Li'iiiueiit  \'iel(l  itleiitical  i'esu]t>.  We 
have,  tlierefore,  apparently,  t^ot  rid  of  the  ermi-  whieli  made 
those  two  ar»iunK'uts  seem  op|)ose(l  to  each  other. 

§  o.  AVe  have  done  moi-e  still.  We  have  Hnallv  reached  a 
universal  formida  such  as  we  lia\c  hcen  scekinu-.      F()r  the  ahove 

.     P..         . 

expression  for     ',  twice  ol)taine(l,  is  a  universal  tornuda  for  the 

constancy  nv  variation  of  the  o-cneral  lexcl  of  prices,  and  the  in- 
verse of  it  is  a  universal  formida  for  the  constancy  or  vai'iation 
of  the  general  exchano-e-vahie  of  monev  in  all  other  thiniis. 

The  above  expression  may  he  simplified,  and,  beino-  re- 
arranged so  as  to  be  brought  into  conformity  with  certain  other 
formulfe,  it  may  lie  written  tlni-. 


And  this  may  be  restated  thus, 

p.._a..+b.+ *' J^' + "' J;?; + 

^'      ^■^''■  + a,    S  +  bJ5  + ■ 

or  thus. 


(1) 


m 


P.,      a., +  b^-|- \    '  -.r,      V    '   - y,,^ 

p^  =  a,  +  b,  -f ■  I    .,,    I    ,,; 

.  la.a.,--  +     (b,b./--  + 


(•5) 


Both  these,  like  the  corresj)onding  formuhe  in  (liapter  X., 
Section  III.  ^  <■>,  namely  those  numbered  (())  and  (7),  are  more 
curious  than  usefid.  Here  we  have  no  additional  form  corre- 
sponding to  formula  (8)  in  that  place.  The  nearest  we  can  get 
to  that  fornnda  is  bv  alterina-  the  last  into  this. 


P.,     a..  +  b.,  + "•^'  M"^"2a, ^ ''■' \'''''^^'b, 

--    -        -  -  .    (4) 


P,      a,  +  b,  + 


I       a..  I       b,, 


382  TlIK    INlVKItSAL    MKTHOI) 

Thus  this  method  is  distinct  from  Scrope's  method  in  any  of  its 
forms  ;  nor  is  it  a  combination  of  any  of  the  forms  of  that 
method. 

If  it  happens  that  the  sums  of  money  spent  on  the  classes  are 
constant,  that  is,  that  x.^a..^  =  x^a^,  yji.^  =  y^^^,  and  so  on,  the 
formulae  reduce  to 

and  consequently  also  to 

Pi      « V.<vr^  -f  [ty  11,11.,  + ' 

which  we  know  to  be  correct  in  these  cases,  as  proved  in  Chap- 
ter X.,  the  latter  being-  Scrope's  method  applied  to  the  geo- 
metric means  of  the  mass-quantities.  ^  If  it  happens  that  the 
mass-quantities  purchased  of  the  classes  are  constant,  that  is, 
that  3^2  =  2/1,  ?/2  =  III,  and  so  on,  so  that  each  may  be  represented 

simply  as  x,  y, ,  or  that  they  have  all  varied  alike  and  x^  = 

ry^,  3/2  =  /7/„  and  so  on,  wherefore,  the  /•  eliminating  itself, 
either  may  be  represented  simply  as  r,  y, ,  the  formulae  re- 
duce to 

P.  _  •'•«2  +  .y/'^2  + 

Pr  ''-^^  +  .'A^,  +  •••'•••  ' 
which  we  know  to  be  correct  for  these  cases,  as  proved  in  the 
last  Chapter,  this  being  Scr<)])e's  method  api)lied  to  the  constant 
mass-quantities.  Thus  the  above  complete  formulae  enclose  both 
those  sets  of  formulae,  just  as  the  universal  conditions  to  which 
the  universal  formulae  are  applicable  enclose  the  two  special  sets 

^  Also  the  same  reduction  will  take  place  if  the  sums  of  money  are  all  in  the 
same  proportion  so  that  a:2«2  =  »'^i"i,  ^2^2  =  »'2/i'^i.  and  so  on,  provided  also  the 
mass-quantities  are  in  the  same  proportion,  that  is,  that  x.,  =  r.Ti,  ijo  =  ri/i,  and  so 
on  ;  but  then  there  are  no  price  variations.  Otherwise,  it  is  only  formulae  (G)  and 
(8)  in  Chapt.  X.  Sect.  III.  jJ  8,  that  are  directly  applicable  in  such  cases,  for- 
mula (8)  remaining  unchanged  (because  •  r  eliminates  itself  from  both  sides  of 

the  fraction),  and  formula  ((>)  merely  reducing  to  a  form  in  which  a,  b,  , 

representing  either  all  the  smaller  or  all  the  larger  sums  spent,  take  tlie  places  of 

aj  and  a2,  to)  and  b^, But  all  the  other  formuhe  are  applicable  under  the 

proviso  that  only  the  sums,  and  only  the  mass-quantities,  of  one  of  the  periods 
be  use<l.     Of.  Note  1  in  Cliapt.  X.  Sect.  1. 


A      I'lIIlM)     MK'rilOI)  .".S,") 

of  eoiiditioiis  to  wliicli  each  of  those  loniiiilii'  was  separately  ap- 
])lical)l('.  . 

§  4.  Desirintr  to  get  clearer  iiisiulit  into  the  meanin<r  of  this 
forimila,  we  may  do  so  by  j)iitting  it  in  another  form,  to  which 
it  easily  reduces,  as  follows, 


P.,  ^","o  ^"-l"-.'  (--,) 


•'■■-  +  .'/■.  Jl        +  "^•'  Ji        + 

■•■, + ^/,  J?# + ^,  i'? + 


The  snI)-numerators  are  the  total  sums  of  money  spent  on  all 
the  goods  at  the  second  and  at  the  Hrst  [)eriods.  The  sub-de- 
nominators are  the  total  numbers  of  mass-units  ])urchased  at  the 
second  and  at  the  first  periods,  that  are  equivalent  over  both  the 
[)eriods  together.  Thus  the  formula  expresses  not  an  average 
of  price  variations,  but  the  variation  (or  constancy)  of  averages — 
to  wit,  the  variation  of  the  average  of  the  })riees  at  the  second 
period  of  the  mass-units  then  purchased  that  are  equivalent  over 
both  the  periods  from  the  average  of  the  prices  at  the  first  period 
of  the  same  mass-units  then  purchased.  These  averages  of  the 
prices  are  arithmetic  averages.  But  the  averages  used  in  ob- 
taining the  equivalence  of  the  mass-units  are  geometric  means. 

Thus  this  method  of  measuring  the  constancy  or  variation  of 
the  exchange-value  of  money  falls  into  line — not  behind  the 
methods  adopted  by  Carli  and  Young,  Jevous,  Laspeyres,  and 
Messedaglia,  and  mostly  employed  hitherto,  of  averaging  merely 
the  variations  of  prices,  nor  behind  the  method  suggested  by 
Scrope,  in  any  of  its  forms,  of  comparing  the  averages  of  prices 
at  each  period  on  the  same  mass-quantities  taken  as  constant  in 
spite  of  facts  to  the  contrary — but  behind  the  method  first  dis- 
covered by  Drobisch,  of  comparing  the  averages  of  prices  at  each 
period  on  the  mass-quantities  of  each  period,  and  so  employing 
what  we  have  called  double  weighting. 

Its  resemblance  to  Drobisch's  method — and  also  its  difterenc(^ 
— is  especially  plain  if  we  suppose  that  the  labor  of  linding  the 


884  THE    rNIVKHSAI,    MKlllon 

mass-units    ('(^uivak'iit   over   l)()tli  the    pt'i-iods  has  already   been 
performed,  and  consequently  the  numbers  of  tliem  purchased  at 

each  period  are  known.    For  tlien,  as  ^^ o-^'o.!'  =  ^ i^y" ,'''■•'  =  i 

formula  (1)  reduces  to 

which,  in  form,  is  the  same  as  the  formula  for  I)rol)iseh"s 
method.  Drobisch,  however,  had  to  use  this  o-oneral  formula  for 
double  weiii'htiuo-  on  the  presu])positiou  that  the  labor  of  obtain- 
ing the  numbers  of  the  mass-units  recommended  bv  him  had 
already  been  performed.  He  was  unable  to  put  iii to  the  general 
formula  itself  his  method  of  selecting  the  mass-units,  or  of  ob- 
taining the  ratio  between  their  numbers.  But  we  have  been  able 
to  do  this  with  our  formula'.  Our  formula,  (1),  for  instance,  is 
applicable  to  the  j)riees  and  numbers  of  any  mass-units  that  hap- 
pen to  be  employed  by  merchants,  as  is  evident  from  the  fact 
that  the  larger  is  the  mass-unit  employed,  the  larger  ^vill  be  its 
prices  and  the  smaller  its  numbers,  and  conversely,  so  that  every 
full  term  (.*'//,,  or  x^Vo.^a.,,  etc.)  remains  unchanged  in  size  what- 
ever be  the  sizes  of  the  mass-units.  What  this  formula  does  is 
to  reduce,  in  its  second  half,  the  numbers  of  the  mass-units  com- 
monly used  to  the  same  proportions  as  are  the  numbers  of  the 
mass-units  e<|uivalent  over  both  the  periods  (as  shown  in  Chap- 
ter X.  Sectiim  III.  §  3),  so  that  we  are  freed  from  the  need  of 
knowing  either  these  mass-units  themselves  or  their  munbers. 

In  the  diffcHMit  mass-units  used,  resulting  in  different  num- 
bers and  pro|)ortions  in  the  second  half  of  the  common  formula 
(6),  lies  the  fundamental  distinction  between  the  present  method 
and  Drobisch's.  Drobisch  sought  to  draw  the  average  |)rice  at 
each  |)eriod  of  the  mass-units,  in  all  the  classes,  that  are  ecpial 
according  to  weight  (or  caj>aeity).  I'liis  method  draAVs  the 
average  price  at  each  period  of  the  mass-units,  in  all  the  classes, 
that  are  e<iual  according  to  exchange-value.  Drobisch's  method 
used  equi])ondcrant  mass-units.  This  method  uses  equivalent 
mass-units — that  is,  ol"  course,  mass-units  that  are  equivalent 
over  both    the   periods  (•om|>ared.      That   this   method    is  more 


A    TIIIIM)    iMKTilOI)  385 

nearly  right  and  tliat  incthod  altogether  wrong,  is  plain  from 
this  distinction  ;  for  it  is  ])lain  that  not  e(|uiponderant,  but  equiv- 
alent, (or  equally  important)  mass-units  are  the  economic  in- 
dividuals the  variations  in  whose  average  price  we  desire  to 
measure. 

From  this  fundamental  distinction  flow  other  differences.  In 
Drobisch's  method  different  classes  can  be  used  at  each  period, 
no  obstacle  being  offered  by  it  against  counting  a  new  class  ap- 
pearing at  the  second  period  any  more  than  against  counting  a 
new  individual  in  an  old  class.  In  the  present  method  only  the 
same  classes  can  be  used  at  each  period  in  the  comparison  of  any 
two  periods  ;  for  otherwise  a  price  quotation  would  be  wanting. 
Thus  this  method  must  obey  the  principles  laid  down  in  Chapter 
IV.  Section  V.  §  9  ;  while  Drobisch's  method  is  free  from  sub- 
jection to  those  principles.  Again,  in  Drobiscii's  method  we 
have  seen  a  grave  defect  to  be  that,  even  though  it  obey  those 
principles,  yet  in  cases  when  between  two  periods  there  are 
irregular  variations  in  the  mass-quantities  but  no  variations  in 
any  prices  whatsoever,  it  may  indicate  a  variation  in  the  general 
exchange-value  of  money,  thus  violating  Propositions  XXVII. 
and  XLIV. ;  and  also  may  give  two  other  indications  which  we 
know  to  be  wrong."  None  of  these  errors  is  committed  by  the 
present  method.  If  no  price  variations  occur,  the  mass-quan- 
tities may  vary  as  they  please,  this  method  indicates  only  con- 
stancy. Or  if  all  prices  vary  in  the  same  proportion,  the  mass- 
quantities  may  vary  as  they  please,  this  method  indicates  only  a 
variation  of  the  general  level  of  prices  in  the  same  proportion. 
This  last  statement,  which  really  embraces  the  first,  may  be 
proved  as  follows.  Suppose  a^  =  ra^,  j3.-^  =  ri9^,  and  so  on,  r 
being  the  common  ratio  of  variation  of  every  price.  (If  there 
are  no  price  variations,  r  =  1.)     Then  formula  (1)  becomes 

P2       Xfa^  -\-  y^rf^^  4- x^ay r  -f  y^f-iyr  + 


■'^i«i  +  yj^x  +  ^''  ( •^•2'^!  +  2/2/^1  + ) 


=  r. 


6  See  Chapt.  V.  Sect.  VI.  I  5. 
25 


386  thp:  universal  :\ietii()d 

whatever  be  tlic  variations  between  x^  and  x.„  y^  and  y^,  etc.  And 
never  will  this  method  indicate  a  rise  of  the  general  level  of 
prices,  when  all  prices  fall,  or  conversfly. 

§  5.  There  is  another  method  which  has  followed  the  general 
lines  of  Drobisch's, — one  which  is  described  near  the  end  of 
A])pendix  C,  but  which  has  not  yet  attracted  our  attention. 
This  is  the  method  invented  by  Professor  Lehr.  In  this  method 
its  author  has  made  an  eifort  to  do  what  appears  to  be  accom- 
plished in  the  method  here  presented.  He  has  tried  to  measure 
the  variation  in  the  average  price  of  mass-units,  in  all  the  classes, 
that  have  the  same  exchange-value  over  both  the  periods  together, 
— to  which  equivalent  mass-units  he  has  given  the  not  inappro- 
])riate  name  of  "  pleasure-units."  The  fault  with  this  method 
lies  in  the  way  it  measures  the  equivalence  of  the  pleasure-units. 
Instead  of  finding  mass-units  between  whose  prices  at  each 
period  the  simple  geometric  means  are  the  same,  it  employs 
mass-units  between  whose  prices  at  each  period  the  unevenly 
weighted  arithmetic  averages  are  the  same.  No  reason  is  assigned 
for  this  choice,  and  it  seems  to  have  been  made  as  a  matter  of 
course.  The  position  is  that  in  a  given  class,  certain  different 
sums  of  money  being  expended  at  each  of  the  two  periods  com- 
pared to  purchase  certain  different  numbers  of  weight-units,  the 
total  of  these  sums  is  expended  over  both  the  periods  together 
to  ])urchase  the  total  of  these  numbers  ;  wherefore  a  single 
money-unit,  on  the  average  over  ])oth  the  periods,  purchases  a 
number  of  weight-units  re])resented  by  the  (|uotient  of  the  total 
sum  of  money  divided  by  the  total  of  the  numbers  of  weight- 
units,  so  that  we  have  here  a  mass  whose  average  price  over 
both  the  periods  is  one  money-unit.  A  similar  operation  is  per- 
formed on  every  class,  in  each  of  which  a  mass  is  obtained 
whose  average  price,  so  measured,  over  both  the  periods,  is  one 
money-unit ;  wherefore  it  is  maintained  that  these  masses,  being 
equivalent  to  the  money-unit  over  both  the  periods,  are  equal 
pleasure-units.  Then  the  rest  of  the  method  is  to  draw  the  aver- 
ages of  the  prices  of  all  these  pleasure-units  of  all  the  classes 
at  each  period,  and  to  compare  them. 

In  this  position  a  first  error  is  the  use  of  uneven  Aveighting. 


A    riniM)  MKTiioi)  387 

If  the  ordiiiarv  coinnu'rcial  mass-unit  of  aiiytliiiiL!;"  be  ])rice(l  at 
one  money-unit  at  tlie  first  period  and  at  two  money-units  at  the 
second,  and  if  the  ordinary  commercial  mass-unit  of  anything 
else  be  priced  at  two  money-units  at  the  first  period  and  at  one 
money-unit  at  the  second,  these  two  mass-units  are,  over  these 
two  periods,  e(|uivalent  mass-units  oi-  equal  ])leasure  units,  with- 
out regard  to  the  number's  of  them  that  may  be  purchased  at 
either  jieriod.  The  numbers  of  them  purchased  at  each  period 
determine  the  relative  importance  of  the  classes,  but  not  tlu>  rcsla- 
tive  importance  of  the  individuals  in  the  classes.  A  consequence 
of  this  error  is  that  undue  influence  may  be  given  to  the  con- 
ditions existing  at  one  of  the  periods  (or  in  one  of  the  countries 
whose  money  is  being  compared  with  another's),  while,  in  truth, 
as  already  })ointed  out  in  Chapter  IV.  (Sect.  V.)  we  ought  to  be 
especially  careful  to  allow  no  greater  influence,  or  weight,  to  one 
of  the  periods  than  to  the  other.  Some  deficiencies  in  the 
method  before  us  following  upon  the  neglect  of  this  principle 
will  be  pointed  out  further  on  in  this  Section  and  later. 

A  second  error  in  this  method  of  obtaining  the  pleasure-units 
is  in  the  use  of  the  arithmetic  average  instead  of  the  geometric. 
In  such  a  measurement  all  the  principles  examined  in  Cha])ter 
VIII.  apply.  If  we  prize  a  mass-unit  of  [B]  twice  as  highly 
as  a  mass-unit  of  [A]  at  the  first  period  and  at  the  second  prize 
the  mass-unit  of  [A]  twice  as  highly  as  the  mass-unit  of  [B], 
it  is  obvious  that  over  these  two  periods  together  we  prize  the 
two  mass-units  equally.  Therefore  the  geometric,  and  not  the 
arithmetic,  mean  is  to  be  used  to  indicate  such  ratios  of  impor- 
tance.^    Also  in  the  arithmetic  averaging  of  the  prices  at  the 

"  Lehr  follows  Drobisch  to  the  extent  of  wanting  us  first  to  reduce  all  mass- 
units  to  the  same  weight-vinit.  This  is  superfluous  in  his  method.  Simple  in- 
spection of  its  formula  will  disclose  that  the  terms  in  it  are  unaffected  whatever 
be  the  size  of  the  mass-units  employed.  It  is  really  a  merit  in  Lehr's  method  that 
it  does  not  I'equire  the  use  of  the  same  weiglit-units). — Another  merit,  wiiere  it 
departs  from  Drobisch's  metliod  and  agrees  witii  the  one  here  presented,  is  that 
it  requires  the  use  of  the  same  classes  at  both  periods.  Wicksell's  criticism  of 
Lehr's  method,  noticed  in  the  Appendix,  thus  strikes  at  a  point  in  it  which 
deserves  credit  instead  of  censure. 

*  Tlie  case  above  cited  would  exist  if  at  tiic  two  jx'riods  respectively  the  prices 
are  :  of  A  1.00  and  1.00,  of  B  2.00  and  .50;  of  A  1.00  and  l.oO,  of  B  2.00  and  .75  ; 
of  A  1.00  and  2.00,  of  B  2.00  and  1.00;  of  A  1.00  and  ..")0,  of  B  2.00  and  .25  ;  or  in 
many  other  coiul)inations.  In  all  these  combinations  tiie  geometric  means  are 
equal ;  the  arithmetic,  only  in  one. 


388  THE    UNIVERSAL    METHOD 

two  periods  a  variation  in  the  exchange-value  of  money  will  de- 
range the  result  of  the  calculation.  But  such  a  variation  has 
no  influence  upon  the  geometric  mean,  as  pointed  out  in  Chapter 
TV.^  It  may  be  added  that  in  using  the  geometric  mean  in  ob- 
taining our  pleasure-units,  we  escape  the  imperfections  in  the 
geometric  average  with  more  than  two  figures,  or  with  uneven 
weighting,  which  are  pointed  out  in  Chapter  VIII.  (Sect.  I.  §  6), 
and  which  have  twice  troubled  us  since,  and  will  trouble  us 
again  presently.  For  here  we  are  using  the  geometric  mean 
proper,  between  only  two  figures,  the  prices  of  the  same  thing 
at  two  periods,  rightly  attaching  equal  importance  to  each  of 
them.  Thus  what  of  the  geometric  method  is  retained  in  our 
final  formula  is  flawless. 

Professor  Lehr's  method  has  the  peculiarity  that  in  consequence 
of  its  merits  and  demerits  just  pointed  out,  it  shares  some  of  the 
defects  of  Drobisch's  method,  and  escapes  others.  Thus  if  no 
prices  vary  between  the  two  periods.  Professor  Lehr's  method 
always  indicates  constancy,  no  matter  what  be  the  variations  in 
the  mass-quantities.  But  if  all  the  prices  vary  uniformly,  this 
method  does  not  necessarily  indicate  the  corresponding  variation 
in  the  general  level  of  prices  unless  all  the  mass-quantities  re- 
main constant  or  also  vary  uniformly."'  Thus,  although,  unlike 
Drobisch's  method,it  respects  Propositions  XXVII.  and  XLIV., 
yet,  like  Drobisch's  method,  it  violates  Propositions  XVII.  and 
XLV.  This  flict  alone  would  be  sufficient  to  show  that  there 
is  somothinu-  wronw;  in  it.  Yet  of  all  methods  hitherto  sim- 
gested  Professor  Lehr's  approaches  the  nearest  to  the  truth  in 
theory,  if  not  also  in  practice. 

^  If  we  retained  this  second,  but  avoided  tlie  first  error,  the  formula  would  be 

Pz  _  a^2"2  +  ^2/^2  + ■Ti(ai+g,)+yi(/3i+/?2)+    

Pi      xia^+yi^i  + '  X2{ai+a2)+y2{fii+$2)+   ' 

This  would  be  better  than  the  method  under  considc>ration,  but  still  not  true. 

1°  If  «2  =  ^''^i,  ^2  —  '■/^i.  ^1*^1  ^'O  on,  Lehr's  formula,  which  is  given  in  Appendix 
C.  VI.  2  -,  reduces  to  this, 

/Xi  +  rx2\         o  (yi+ry2\, 

P2      r(x2a,+y2l3,  + )      ^^"^  x^  +  x J  +  ^^^ A  yi+y^)^ 

Pi         ^i«i  +.Vi/?i  + .^,a  /^i +1^2 W      ^  (^t+Z^\  ^ • 

-  ^\  X1+X2  J     ^    ^\  2/1  +3/2  ^ 
This  reduces  to  unity  if  r--  1 ;  but  otherwise  it  reduces  to  r  only  if  X2  =  3Xi, 
2/2  —  ■'•'2/1)  ^'1^1  so  on. 


A  Tiiiiu)  mi:tii()])  .'>'S9 

§  (J.  There  is  still  another  way  in  which  these  three  methods, 
which  use  double  weighting,  may  be  compared.  This  is  by 
comparing  the  relations  of  their  results  in  a  series  of  periods. 
In  any  such  series  we  know  with  certainty  that  the  results  ob- 
tained serially  ought  to  agree  with  the  results  obtained  directly  : 
(1)  if  the  prices  have  all  remained  constant  through  the  whole 
series,  or  have  all  varied  alike  at  any  or  every  stage,  no  matter 
what  be  the  changes  in  the  mass-quautities  ;  (2)  if  the  mass- 
quantities  have  all  remained  constant  throughout,  or  have  all 
varied  alike  at  any  stage,  no  matter  what  be  the  changes  in  the 
prices  ;  (3')  if  at  all  consecutive  periods  except  two  the  states  of 
things  are  exactly  the  same,  or  the  variations  are  all  in  the 
same  proportion,  this  having  the  eifect  of  reducing  the  irregular 
variations  in  the  series  to  two  sets  ;  and  (4)  if  the  state  of  things 
at  the  last  period  is  exactly  the  same  as  at  the  first,  or  even 
diiferent  always  in  the  same  proportion,  no  matter  what  in- 
tervening changes  have  occurred.  Or  even,  if  wc  grant  Pro- 
fessor Westergaard's  position,  we  may  dispense  with  all  these 
restrictions,  and  say  that  the  agreement  ought  to  take  place  in 
any  and  every  possible  case,  no  matter  what  be  the  changes  in 
the  prices  and  in  the  mass-(iuantities. 

To  begin  with  a  series  of  three  periods  :  taking  the  formula 

(1)  above  reached  for  p",  and  similarly  framing  the  formula  for 

p 

p-^ ,  and  multiplying  these  by  each  other,  we  get  the  formula  for 

the  method  here  presented  as  it  serially  indicates  the  variation 

/       P    P       P  \ 

from  the  first  to  the  third  period  (  ^^r  p-  •  p-  =  p^  j  .    This  for- 
mula we  find  to  be 


•^s^s  +  2/3/^3  4- 


( a;y^,+  yy;3A  + )  ( xy^+yyMs±-_:::::) _ 

( .ry^,+yy]gi,+ )  ( :ry«^,+?/y/?,/33-H )■ 

But  the  formula  for  the  direct  comparison  of  the  third  with  the 
first  period  is 


390  THE    UNIVERSAL    METHOD 

Pj  _  -I's^s  +  yA  + a-yafy  +  yy(iu%  + 


These  formulae  may  agree  by  chance  if  it  happens  that  in  the 
former  the  Uist  two  thirds  yield  products  in  the  numerator  and 
in  the  denominator  equal  to,  or  in  the  same  ratio  as,  the  numer- 
ator and  the  denominator  in  the  last  half  of  the  latter.  They  will 
regularly  agree,  as  is  easily  perceived :  (1)  if  «3  =  'z.,  =  «i,  and  so 
on  with  all  the  prices,  or  if  a^  =  ra^  and  a.,  =  sa^,  and  all  the 
other  price  variations  be  in  the  same  ratios,  that  is,  if  there  be  no 
price  variations,  or  if  all  the  price  variations  at  each  stage  be  in 
the  same  ratio,  no  matter  what  be  the  variations  in  the  mass- 

quantities  I  in  the  former  case  ^^  =  1.00,  in  the  latter ^^  =  r  s\; 

(2)  if  x^  =  .1*2  =  .i\  and  so  on,  or  if  x^  =  rx.^  and  x^  =  sx^,  and 
so  on,  that  is,  if  there  be  no  variations  in  the  mass-quantitiesj 
or  if  all  such  variations  at  each  stage  be  in  the  same  ratio, 
no  matter  what  be  the  price  variations  ;  (3)  if  r/^  =  «^,  or  a^  =  ra^, 
and  so  on,  provided  either  x^  =  x^  or  x^  =  s  x^  and  so  on,  or  if 
"a  =  ^^2'  ^^  ^3  =  '"^'^'"  ^^^  ^^  ^^^?  provided  either  x^  =  x.^  or  iv^  = 
sx^  and  so  on,  that  is,  if  there  be  only  one  stage  with  irregular 
variations  ;  (4)  if  a^  =  a^  and  so  on,  provided  either  x^  =  x^  or 
x^  =  scCj  and  so  on,  in  which  cases  both  the  formulge  give  unity  for 
result,  indicating  sameness  of  the  price-level,  or  if  a^  =  ra^  and 
so  on,  under  either  of  the  same  provisos,  in  which  case  both  the 
formuhc  give  r  for  result,  indicating  a  general  price  variation  the 
same  as  all  the  particular  price  variations,  the  conditions  here 
being  merely  that  there  be  no  irregular  differences  between  the 
first  and  tlie  last  periods,  no  matter  what  intervening  changes 
may  have  taken  2)lace  ;  but  in  no  other  cases,  tliat  is,  not  univer- 
sally or  unconditionally. 

Examining  Drobisch's  formula  in  tlie  same  way,  we  find  that 
the  two  measurements  universally  and  unconditionally  agree. 
Thus  Drobisch's  method  completely  satisfies  all  these  tests. 
This  fact  we  have  already  noticed ;  but  we  have  seen  that  the 
correctness  of  Drobisch's  method  cannot  thereby  be  proved.  It 
shares  this  advantage  with  other  methods  clearly  fiilse.  And  it 
fails  before  other  simpler  tests. 


A    THIRD    METHOD  391 

Again,  exainiiiing  Professor  Lelir's  method  in  the  same  way/^ 
we  find  that  the  two  measurements  agree  in  cases  restricted  to 
half  of  each  of  the  four  divisions  in  which  the  method  here  ad- 
vanced is  consistent.  They  agree  only  (1)  if  a^=  a^  =  a^  and  so 
on,  that  is,  if  there  be  no  price  variations  at  all,  no  matter  what 
be  the  variations  in  the  mass-quantities  ;  (2)  if  x^  =  x^  =  x^  and 
so  on,  that  is,  if  the  mass-([uantitics  be  constant,  no  matter  what 
be  the  price  variations  ;  (3)  if  a^=  a^  aud  so  on,  provided  x^=  x^ 
and  so  on,  or  if  a.^  =  a.^  and  so  on,  provided  x^  =  x.^  and  so  on, 
that  is,  if  there  be  only  one  stage  with  any  variations  at  all ;  (4) 
if  «g  =  «j  and  so  on,  provided  x^  =  x^  and  so  on,  that  is,  if  the 
states  of  things  be  exactly  the  same  at  the  last  as  at  the  first 
period,  no  matter  what  be  the  intervening  changes.  It  will  be 
observed  that  in  every  one  of  the  four  divisions  the  proportional 
.variations  are  excluded. 

There  is  still  another  possible  method  using  double  weight- 
ing, which  deserves  to  be  noticed  here  for  the  sake  of  complete- 
ness. This  is  a  form  in  which  Professsor  Nicholson's  method 
admits  of  being  stated  (the  third  form  given  in  Appendix  C, 
VI.  §3).  A  general  reason  for  its  not  being  a  successful  method 
is  that  it  uses  even  weighting  for  the  inverted  variations  of  the 
mass-quantities.  In  particular,  it  falls  most  abjectly  before 
these  tests.  The  agreement  between  its  two  measurements  de- 
pends entirely  upon  the  behavior  of  the  mass-quantities.  It 
takes  place  only  if  .r.,  ^  .r^,  or  a?.,  =  sx^,  aud  so  on,  or  if  x^  =  .r„ 
or  .1*3  =  rx.^,  and  so  on,  that  is,  only  if  there  be  no  irregular  varia- 
tions in  the  mass-quantities  at 'least  in  one  of  the  stages.'^ 

p 

11  Or  it  may  he  easier  to  examine  it  iu  conformity  with  this  formula:  ^  = 

Pi         Pi" 

12  If  3^3  =  Xi  and  so  on,  .i-j  being  different,  the  arithmetic  average  of  the  vari- 
ations of  the  mass-quantities  (in  the  last  half  of  the  formula)  becomes  in  the 
indirect  comparison,  a  harmonic  average  of  them,  likewise  with  even  weighting. 
— The  method  there  also  (in  Note  4)  suggested  as  a  variant  (with  the  geometric 
average  of  the  inverted  variations  of  the  mass-quantities)  has  the  merit  that  it 
universally  satisfies  Westergaard's  test;  but  it  likewise  has  the  defect  of  using 
even  weighting  in  averaging  the  inverted  variations.  If  we  cui-ed  it  of  the  latter 
defect  by  using  uneven  weighting  adapted  for  every  comparison  (as  there  formu- 
lated in  Note  5)  it  would  again  lose  the  former  merit. — The  method  above  sug- 
gested in  Note  9,  sul)mitted  to  these  tests,  yields  results  that  agree  iu  exactly  the 
same  cases  as  Lehr's  method. 


392  THK    UNIVERSAL    METHOD 

§  7.  In  a  series  with  four  or  more  periods  the  employment  of 
these  tests  becomes  too  cumbersome  to  make  it  worth  while  to 
pursue  this  enquiry  into  much  detail.  We  must,  however,  ex- 
amine what  happens  in  such  series  to  the  metliod  here  advanced. 
The  indirect  comparison  of  the  fourth  period  witli  the  first,  very 
much  abbreviated,  is  as  follows  : 

P4^^'4"4+ {xya^a^-i- )   (a;aT/»;a34- )  {xyaia^-\- ^ 

^1       ^'i°i+ {x^V"^,-\- )    (.r3l/^3+ ){x,Va,a,  + ), 

while  the  direct  comparison,  equally  abbreviated,  is 

P4 a\ai  + ^i^'*i«i+ •••::: 

i'i~~«i«i  + '  xy^^-i- ' 

These  will  regularly  agree  (1)  if  a^  =  a^=  a,  =  a^ ,  or,  more  com- 
prehensively, if  a^  =  ra^ ,  a^  =  sa.^  and  a.^  =  ta^  ,  and  so  on  in 
every  case,  that  is,  if  there  be  no  price  variations,  or  if  all  the 
price  variations  at  every  stage  be  in  the  same  ratio,  no  matter 
what  be  the  variations  in  the  mass-quantities ;  (2)  if  .r  ^=  x^  = 
x.^  =  .Tj ,  or  if  x^  =  rx^ ,  x.^  =  sx.,  and  x.^  =  tx^ ,  and  so  on  in  every 
case,  that  is,  if  there  be  no  variations  in  the  mass-quantities,  or 
all  such  variations  at  every  stage  be  in  the  same  ratio,  no 
matter  what  be  the  variations  in  the  prices ;  (3)  if  irregular 
variations  take  place  only  at  one  stage,  all  the  periods  then  vir- 
tually reducing  to  two.''  And  it  is  evident  that  the  series  may 
be  extended  to  any  length,  exactly  these  relations  will  hold — 
the  two  comjjarisons  will  regularly  agree  if  any  of  these  condi- 
tions be  observed  ;  but  not  necessarily  in  other  cases. 

Now  among  the  cases  in  which  these  comparisons  will  not 
necessarily  agree  is  the  one  in  which  everything  at  the  last  period 
is  exactly  what  it  was  at  the  first  period — both  all  the  prices  and 
all  the  mass-quantities.  In  this  case  we  know  with  absolute 
certainty  that  the  two  comparisons  ought  to  agree,  no  matter 
what  be  the  intervening  changes.  Our  new  method,  therefore, 
may  lead  to  error.  But  among  the  methods  we  have  been  re- 
viewing it  is  only  Drobisch's  method  (along  with  a  few  other 
false  metliods  not  here  noticed  '^)  that  still  holds  out  against  this 
test. 

^^For  Lehr's  method,  and  its  modification  suggested  in  Note  9,  there  are  tlie 
same  conditions  with  exclusion,  as  before,  of  the  proportional  variations. 

^•^  i?.  g.  the  method  ahove  alluded  to,  suggested  in  Appendix  C,  V.  §3,  Note  4. 


A    THIRD    METHOD  393 

§  8.  Thus  our  method  fails  us  twice.  It  fails  even  in  a  series 
of  three  periods  to  satisfy  Professor  AVestergaard's  full  test, 
although  in  that  series  it  satisfies  the  certain  test  yielded  by  sup- 
posing an  exact  reversion.  But  in  a  longer  series  it  does  not 
even  satisfy  this  latter  test.  It  behooves  us  then  to  advert  to 
this  defect  in  it. 

This  defect  in  the  complete  method  is  obviously  a  survival  of 
the  defect  in  the  method  for  constant  sums  of  money,  above 
examined  at  the  end  of  Chapter  X.  The  complete  method  we 
started  out  to  construct  upon  each  of  the  partial  methods,  by 
extending  and  by  modifying  them.  But  it  turns  out  that  it 
modifies  much  more  the  method  for  constant  mass-quantities 
than  the  method  for  constant  sums.  Now  it  was  precisely  the 
method  for  constant  mass-quantities  that  was  perfect ;  and  this 
has  been  modified,  while  the  method  for  constant  sums  has  been 
incorporated  whole  in  half  of  every  formula  for  our  complete 
method.  Hence  the  imperfection  in  the  method  for  constant 
sums  has  come  over  entire  into  this  complete  method. 

In  that  method  we  discovered  the  cause  of  the  defect,  and  a 
way  of  getting  rid  of  it,  though  not  without  loss  of  other  qual- 
ities. The  defect  can  be  dispelled  by  taking  the  numbers  at 
each  period  of  the  mass-units  that  are  equivalent  over  all  the 
periods — the  pleasure-units,  as  Professor  Lehr  calls  them,  not 
of  two  periods  at  a  time,  but  of  the  whole  comprehensive  epoch. 
Having  already  W'Orked  these  out,  we  should  have  the  following 
formula  for  our  complete  method,  expressed  in  our  usual 
symbols,^"* 

P,     a^  +  b,  +  ■  x^"  +y^"  + ■  ^^ 

And  now,  as  all  the  terms  remain  the  same  for  every  period  in 
all  the  comparisons,  the  measurements  wall  all  agree  whether 
made  directly  between  any  two  distant  periods  or  serially  through 
the  intervening  periods.  Here,  then,  we  have  an  at  least  con- 
sistent method. 

But  the  alteration  we  saw  to  be  impracticable  in  the  case  of 
the  partial  method.      It  is  equally  impracticable  for  the  com- 

^^  Cf.  the  formula  in  Chapt.  X.  Sect.  Y.  §  7. 


394  THE    UNIVERSAL    METHOD 

pletc  method.  Wo  arc  obliged,  therefore,  to  get  along  with  a 
slightly  imperfect  form  of  the  complete  method — which,  how- 
ever, we  shall  find  reason  to  believe  no  worse  than  if  it  were 
revised  in  this  way. 

In  the  earlier  Chapter  we  examined  the  probable  amonnt  of 
error  incnrred  by  using  this  defective  feature  in  the  partial 
method.  We  found  it  very  small  for  ordinary  cases ;  and  also 
we  detected  in  the  method  conflicting  and  neutralizing  tendencies, 
so  that  when  we  deal  with  many  classes,  and  in  a  long  series, 
the  i)robabilities  are  that  the  results  will  never  deviate  to  any 
great  extent  on  either  side  of  the  truth,  but  that  they  will  pass 
from  the  one  side  to  the  other  of  it,  always  keeping  it  close 
company,  and  often  coinciding  with  it.  Now  the  defective 
feature  in  the  whole  of  that  partial  method,  which  here  forms 
onlv  half  of  the  complete  method,  enters  into  a  composition  in 
which  the  other  half  is  without  defect.  Hence  it  might  seem  as 
if  tlie  error  incurred  through  that  defectiveness  would  be  diluted, 
and  lessened  by  half.  Unfortunately  this  is  not  so.  The  de- 
fectiveness of  the  half  leavens  the  whole.  Still  the  error  cannot 
be  greater  here  than  there.  The  error  being  the  same,  we  need 
not  examine  it  again. 

§  9.  The  defect  which  has  been  pointed  out  in  tin's  method 
belongs  to  it  in  a  series  of  years,  the  reason  for  its  existence 
there  beina:  obvious.  This  reason  for  the  defect  does  not  touch 
the  method  used  in  making  a  direct  comparison  between  two 
periods — especially  between  two  contiguous  periods.  The  (ques- 
tion still  remains  :  In  comparing  two  periods  is  this  a  perfect 
method? — is  it  the  one  true  method?  The  reasoning  by  whicli 
it  has  been  reached  seems  to  be  faultless.  Yet  there  was  a  fatdt 
in  one  of  the  premises — in  one  of  the  partial  methods.  And  the 
method,  even  in  comparing  two  periods,  is  not  perfect.  We 
have  seen  it  stand  certain  tests.  Unfortunately  it  does  not  stand 
all  tests. 

One  of  the  tests  to  whicli  we  subjected  the  ])artial  methods 
pr(3viously  reached  for  cases  witli  constant  money-sums  or  with 
constant  mass-quantities,  was  examination  as  to  whether  they 
carried  out  Pr()|)osition  XXXVI.,  or  not.     This,  it  may  be  re- 


A    THIRD    METHOD  .395 

called,  is  the  jierfectly  evident  [)riMeij)le  that  if,  luakiiig  a  calcu- 
lation u})<)n  all  but  one  or  more  classes,  we  find  the  general 
price  variation  to  be  indicated  at  a  certain  ratio,  and  if,  later 
noticing  the  other  class  or  classes,  and  finding  them  all  to  vary 
in  price  exactly  in  this  ratio,  we  insert  them  in  the  calculation, 
which  we  ])erf()rni  over  again,  the  result  yielded  in  the  later  re- 
calculation ought  to  agree  with  the  result  first  obtained.  Now 
we  have  seen  that  the  present  complete  method  is  composed  of 
those  two  incomplete  methods  ;  and  we  previously  saw  that  each 
of  those  methods  stood  this  test,  provided  its  own  conditions 
were  observed,  and  not  otherwise/^  Those  conditions  cannot 
exist  together  (except  in  absence  of  all  variations) ;  and  we  are 
now  investigating  cases  in  which  both  are  supposed  to  be  broken. 
We  have  seen,  also,  and  see,  that  that  Proposition  applies  to  all 
possible  cases,  no  matter  what  may  be  the  money-values  of  the 
classes,  or  their  mass-quantities.  Does,  then,  our  present  method 
satisfy  this  test  ?     It  does  not. 

That  it  does  not  is  seen  most  easily  by  taking  formula  (5)  and 
treating  it  in  the  way  supposed.  If  in  all  but  one  class  the  in- 
dication is  of  a  variation  in  the  general  level  of  prices  from  1 
to  r,  it  must  be  that 

a^2^«i«2 + 2/2  ^M + \  -^-y^i^o + 2/yM  + / ' 

and  now  if  j%  =  ry^,  it  ought  still  to  be  that 

a«i  +  VA  +  +  Vi 


'  V  xy^.^  +  y,<v,  + +  z.ry'r)' 


But  this  is  not  necessarily  true, — nor  is  it  necessarily  true  even  if 
z.,  =  Ji'j ;  nor  even  if  /•  =  1  ;  nor  even  if  both  these  conditions 
occur  together  (unless  the  numerators  themselves,  the  total  sums 

^''Except  in  oiu-  form  of  tlieir  formulae,  common  to  both,  namely  Scrope's 
method  applied  to  the  geometric  means  of  tiieir  mass-quantities,  which  we  have 
called  Scrope's  emended  method,  and  from  which  the  present  method  has  been 
distinguished. 


396  THE    1TNIVERSAI>    ^fETHOD  J 

spent  at  eacli  peruKl,  are  e(|ual).  And  the  same  rcsnlt  is  ob- 
tained by  using  any  of  the  otlier  fornudie  for  this  method.'" 

Here,  then,  in  this  method  is  a  grave  theoretical  defect.  It 
Avill  hardly  lead  to  any  inconsistency  in  practice,  since  we  are 
not  apt  to  make  such  recalculations,  nor  are  such  coincidences 
likely  to  be  found.  But  the  existence  of  this  theoretical  defect 
shows  that  the  method,  even  in  a  single  comparison  between  two 
contiguous  periods,  is  not  perfect. 

This  defect  exists  in  all  the  other  methods  using  double 
weighting — including  Drobisch's.^**  It  is  inherent  in  all  methods 
using  double  weighting,  no  matter  which  of  the  three  kinds  of 
averages  of  the  prices  be  used.  But  from  it  are  totally  free  all 
methods  using  single  weighting  (averaging  the  price  variations),'^ 
and  Scrope's  method  in  all  its  forms  (this  being  reducible  to  the 
preceding). 

Hence  in  our  search  after  perfection  we  are  thrown  back  upon 
these  older  styles,  though  upon  methods  never  before  employed 
or  suggested,  and  must  examine  whether  they  are  better. 

III. 

§  1.  That  both  the  geometric  method  and  Scrope's  emended 
method  carry  out  Proposition  XXXVI.  is  evident  upon  simple 
inspection  of  their  formulae.  They  thus  avoid  one  of  the  de- 
fects in  the  method  with  double  weighting  above  investigated. 
We  now  need  to  examine  whether  they  avoid  the  other  defects 
in  that  method. 

The  geometric  method  does  not  universally  satisfy  Professor 
Westergaard's  test,  as  we  already  know.'  It  satisfies  it,  as  is 
easily  perceived,  in  a  scries  of  three  periods  only  under  tliree  of 
the  four  full  conditions  that  we  have  seen  to  be  required  for  the 
method  witli  double  weiy-liting;,  and  in  a  lons>:er  series  onlv  under 
two  of  tlie  three  there  allowed.     It  satisfies  it  under  the  first  of 

i'  Its  failure  in  tliis  respect  leads  to  a  defV'et  in  this  method  when  extended  to 
measuring  eonstaney  (or  variation)  in  oxehange-value  in  all  things,  which  will 
be  noticed  in  the  next  Chapter. 

''  And  the  method  suggested  in  Appendix  C,  §  3,  Note  4. 

"Cf.  Appendix  A,  I.  ?  8. 

'  See  Chapt.  V.  Sect.  VI.  ^  7. 


COMPARISON    OF    THE    THREE    METHODS  397 

those  conditions,  l)ecause  then  the  weighting  is  indifferent ;  ^ 
under  the  third,  because  all  methods  do  so  under  that  condition  ; 
and,  in  the  former  case,  under  the  fourth,  because  then  the 
weighting  is  the  same  in  both  the  comparisons.  But  it  does 
not  satisfy  it  under  the  second  condition,  in  either  case  ;  for 
here  there  may  be  price  variations  with  different  weighting  in 
the  comparisons.  Thus  this  method  does  not  behave  quite  so 
well  as  that  method.  Like  it,  it  fails  in  a  long  series  also  in 
the  certain  case  where  sameness  of  the  price-level  should  be 
shown  at  the  end  of  the  serial  calculations,  when  everything  at 
the  last  period  returns  to  what  it  was  at  the  first.  We  have 
seen,  however,  that  all  such  inconsistency  would  cease,  if  the 
same  weighting  be  employed  through  the  whole  epoch,  such 
weighting  properly  being  the  geometric  average  of  the  full 
money-values  of  the  classes  at  every  one  of  the  periods  in  the 
epoch.  Bat  we  have  objected  to  such  a  procedure  as  being  less 
trustworthy  than  the  use  of  its  own  weighting  in  every  com- 
parison of  two  periods. 

Scrope's  emended  method  fails  also  before  this  test,  but  only 
as  did  the  method  with  double  weighting.  In  a  series  of  three 
periods,  its  indirect  comparison  of  the  third  with  the  first  is  as 
follows  : 


3 


Pi       {ayx^x.,  +  i^yiuy.  -f )  {"-y^-fi  +  l^2^i/-2!f-,  + ) 

while  the  direct  comparison  is 

^3_  «3^^3  +  l^yy^ys  + 

Pi    ayx.x^  -f-  /9y 3/12/3  + ' 

It  is  plain  that  these  regularly  agree  only  under  all  the  four  full 
conditions  above  noticed  for  the  method  with  double  weighting. 
These  may  be  briefly  recapitulated.  They  are  :  (1)  if  ^^  =  ra^ 
=  r.s'«|  and  so  on  ;  (2)  if  x^  =  r.v.,  =r.'<x^  and  so  on  ;  (3)  if  a^ 
=  ra^,  provided  x^  =  sx^,  and  so  on,  or  if  a^  =  va.,,  j)rovided  x^ 
=  .s'.r.,,  and  so  on  ;  (4)  if  a^  =  ra^,  provided  x^  =''^-t*i,  imd  so  on. 
^  See  Appendix  A,  I.  \  8. 


398  THE    UNIVERSAL    METHOD 

And  in  a  series  of  four  or  more  periods,  its  indirect  comparison 
of  the  fourth  with  the  first  is  as  follows,  abbreviated  as  before : 

P^  ^  (ayx^x,  + )  {o..yx.^x.^  + )  (ayx^x^  + ) 

1^        {ayx^x^  + )  («X-«2-^3  + )  («3^"'3^4  + )' 

wliih'  the  direct  comparison  is 

r*4      fy.yx^x^  4  

Pi        ayx^x^  +  

And  again  it  is  evident  that  these  regularly  agree  only  under 
the  three  conditions  noticed  for  that  method,  namely  (1)  if  a^ 
=  ya^  =  ys«2  =  rsta^  and  so  on  ;  (2)  if  x^  =  rx^  =  r.s.r.,  =  rdx^^ 
and  so  on  ;  (3)  if  there  be  irregular  variations  only  at  one  stage. 
The  series  may  be  extended  indefinitely  :  the  formulae  and  the 
conditions  Avill  merely  be  extensions  of  these. 

It  is  plain,  again,  that  also  this  emended  form  of  Scrope's 
method  can  be  still  further  emended  and  cured  of  this  defect, 
if,  instead  of  applying  it  to  the  geometric  means  of  the  mass- 
quantities  at  each  of  the  two  periods  in  every  comparison,  we 
apply  it  in  every  comparison  to  the  geometric  average  of  the 
mass-quantities  over  the  whole  epoch. 

This  is  a  remedy  similar  to  those  we  had  to  invoke  for 
the  other  two  methods.  To  review  :  The  geometric  method 
of  averaging  price  variations  (with  single  weighting)  can  be 
made  universally  to  satisfy  Professor  Westergaard's  test  by 
using  it  with  the  same  weighting  over  the  whole  epoch, — say 
of  n'  periods, — that  is,  with  weights  that  are  the  geometric 
averages  of  the  sums  spent  on  every  class  at  every  pei;iod  (e.  g. 

V-^/Zj  "^^S ^  ^^'  terms  ).  The  method  with  doul)le  weigh- 
ting above  expounded  can  be  made  universally  to  satisfy  that  test 
by  using  it  with  mass-units  e(piivalcnt  over  the  whole  epoch,  as 
found  by  geometrically  averaging  the  prices  of  the  ordinary  mass- 
units  in  every  class  at  every  period  (cf/.  \^''o.^  ■  a^  ■ to  n'  terms). 

And  now  the  emended  form  of  Scrope's  method  can  be  made 
universally  to  satisfy  that  test  by  applying  it  to  mass-quantities 
that  are  the  geometric  average  of  the  mass-quantities  of  every 


COMPARISON    OF    THE    THREE    METHODS  399 


class   at   every  |)eriod    (e.  //.  '^^^x^-x.^ to  n'  terms).      It   is 

therefore  incumbent  upon  us  to  examine  this  system  of  further 
emending-  and  revising  the  three  methods  for  the  sake  of  remedy- 
ing one  of  their  defects — in  the  last  its  only  known  defect. 

§  2.  In  no  case  is  this  remedy  satisfactory,  for  two  principal 
reasons: — (1)  Because  the  present  epoch  is  extending  every 
year,  requiring  recalculations ;  and  it  does  not  appear  that  a 
later  recalculation  will  be  more  correct  than  an  earlier.  Besides, 
how  is  a  past  variation  between  two  years  several  years  ago  to 
be  affected  by  present  variations  ?  (2)  Because  we  really  do  not 
know  how  to  calculate  weights,  or  to  determine  equivalence  of 
mass-units,  or  to  average  mass-quantities,  over  more  than  two 
periods,  since  the  geometric  average  loses  its  virtue  when  aj)- 
plied  to  more  than  two  figures.  Hence  it  may  be  that  in  work- 
ing over  these  methods  into  methods  universally  satisfying 
Professor  Westergaard's  test  Ave  gain  consistency  between  cross- 
measurements  at  the  expense  of  other  qualities. 

This,  in  fact,  can  be  shown  to  be  the  case  with  the  method 
using  double  weighting.  Suppose  states  of  things  over  three 
periods  as  follows  : 

I  100  A  @  1.00     100  B  @  1.00      100       for  [A]     100  for  [B], 

II     75  A  @  1.50     130  B@    .60  I  112.50  for  [A]       78  for  [B], 

III  100  A  @  1.00     120  B@  1.00  I  100       for  [A]     120  for  [B]. 

Going  from  period  to  period  the  method  with  double  weighting 

P  P  P 

gives  these  results  :  p'  =  0.9890  and  ^^  =  1.0322,    whence  ^ 

^  1.0208,  which  is  2.08  per  cent,  above  what  we  take  to  be  the 
true  position.  In  the  direct  comparison  of  the  third  period  with 
the  first  this  method  rightly  indicates  sameness.  Now  when 
this  method  is  worked  over  so  as  to  avoid  this  inconsistency,  it 

P  P  P 

gives  these  results  :     -  =  ().i»(j8G  and  p=  l.U4()S,   whence  p"^ 

^1  ■'-2  ^1 

=  1.0139  ;  and  in   the  direct  comparison  of  the  third  with  the 

P 

first   period   this   method    still    indicates  -p'^  ^  1.01.39.      Thus, 

^  1 

though  consistent,  the  revised  method  twice  gives  a  \vrong  re- 
sult, being  1.39  per  cent,  too  high. 


400  THE    UNIVERSAL    METHOD 

On  this  schema  the  two  other  methods,  going  from  period 
to  period,  give  results  very  close  to  those  given  by  the 
method    with  double  weighting  similarly  used.     The  geometric 

P  P  P 

method   yields    p'-  =  0.9902  and  ^^  =  1.0310,  whence -p^' = 

P 

1.0209.     And    Scrope's    emended    method  yields   p-  =  0.9885 

-•-1 

P  P 

and  -p^=  1.0325,    whence  p^  =  1.020G.     But  when  they  are 

worked  over  to  cover  this  .short  epoch  of  three  periods,  in  avoid- 
ing inconsistency  in  their  final  results,  they  also  avoid  the 
error   incurred   by  the  preceding  method.       For,  so   used,    the 

P  P 

geometric  method  yields  p"  =0.9698  and  ^  =  1.0311,  whence 

P 

p^  =  1 ,00,  which  is  also   indicated  in  the   direct  comparison ; 

.  P  P 

and  Scrope's  emended  method  yields  p-  =  0.9953  and  p-  = 

1  2 

P 

1.0046,  whence  p'^  =  1.00,    which    is  also  indicated  in  the  di_ 

rect  comparison. 

The  question,  however,  arises  :  Is  Professor  Westergaard's 
test  correct  universally  ?  The  case  before  us  is  of  such  a  nature 
as  to  throw  doubt  upon  it.  Here  the  prices  of  both  the  classes 
and  the  mass-quantity  of  [A]  alone  have  reverted  at  the  third 
})eriod  to  what  tlicy  were  at  the  first.  Had  this  third  period, 
witli  the  sameness  of  its  prices,  immediately  followed  u])()n  the 
first  period,  there  would  be  no  question  but  that  the  exchange- 
value  of  money  is  constant  (in  accordance  with  Proposition 
XLIV.).  But  an  intervening  period  has  separated  the  two  ; 
and  now,  while  the  same  number  of  economitMudividuals  in  [A] 
fall  in  price  during  the  second  stage  as  rose  during  the  first,  in 
[B]  a  greater  number  of  such  individuals  rose  in  price  during 
the  second  stage  than  fell  during  the  first.  Had  the  mass-quan- 
tity of  [B]  fallen  back  at  the  third  period  to  what  it  was  at  the 
first,  there  could  again  })e  no  question  but  that  the  level  of 
prices  at  the  third  period  has  returned  to  what  it  was  at  the  first, 


COMPARISON    OF    THE    THREP:    METHODS  401 

— which  would  be  indicated  by  all  these,  and  by  several  other, 
methods.  But  ought  not  the  fact  that,  while  the  changes  in  [A] 
counterbalance  each  other,  a  greater  number  of  individuals  in 
[B]  have  risen  than  fallen,  be  allowed  to  show  that  the  level  of 
prices  has  risen  more  than  it  has  fallen,  so  that  it  is  rightly 
placed  by  all  our  three  methods,  in  their  serial  use,  slightly 
above  its  first  position  ?  This  would  involve  also  that  the  ex- 
change-value of  money  has  fallen  somewhat, — which,  seeing  that 
prices  are  exactly  the  same,  is  somewhat  hard  to  entertain. 

Yet  another  example  casts  doubt  upon  Professor  Wester- 
gaard's  universal  test.  Su])pose  that  both  prices  and  mass- 
quantities  vary  irregularly  between  a  first  and  a  second  period. 
And  suppose  that  between  the  second  and  a  third  period  there 
is  irregular  variation  of  mass-quantities,  but  no  variation  of 
prices.  There  is,  then,  no  variation  of  the  general  price  level 
between  these  last  periods.  Therefore,  the  indirect  comparison 
of  the  third  with  the  first  period  will  show  the  same  general 
price  variation  as  between  the  second  and  the  first.  But  the 
direct  comparison  of  the  third  with  the  first  will  show  a  differ- 
ent price  variation  from  that  between  the  second  and  the  first, 
and  consequently  from  that  indirectly  obtained  between  the 
third  and  the  first.  Xow  of  these  two  measurements  the  latter 
has  more  reason  in  its  favor. 

Still,  even  if  we  should  deny  Professor  Westergaard's  test 
in  such  cases,  we  should  gain  little  comfort  in  regard  to  our 
methods,  since  there  is  one  case  in  which  his  test  is  perfectly 
certain,  and  which  none  of  the  methods  (except  in  their  doubt- 
ful revised  forms)  can  satisfy.  This  is  when  at  any  later  period 
the  prices  and  the  mass-quantities  both  revert  exactly  (or  pro- 
portionally) to  what  they  were  at  some  earlier  period.  This  is 
a  test  which  no  sound  method  yet  devised  or  suggested,  in 
going  from  period  to  period  over  all  the  intervening  periods, 
will  stand. 

That  is,  none  of  the  known  methods  that  hold  out  against 

other  tests  will  stand  this  test  in  the  world  such  as  we  have  it 

— with  varying  mass-quantities  as  well  as  with  varying  prices. 

But  if  we  had  an  economic  world, — or  supposed  one, — in  which 

26 


402  THE    UNIVERSAT.    :S[ETHOD 

forever  the  same  mass-quantities,  or  mass-quantities  in  the  same 
proportions,  are  bought  and  sold,  at  prices  varying  according 
to  demand  only,  then  both  Scrope's  emended  method  and  the 
method  with  double  weighting  above  described — both  of  which 
are  in  these  cases  the  same  as  Scrope's  method  applied  to  the 
constant  mass-quantities, — w(ndd  completely  and  absolutely  sat- 
isfy Professor  Westergaard's  test,  and  all  other  tests.  In  such 
a  world,  the  argument  for  this  method  being  convincing,  we  can 
be  certain  that  we  have  the  absolutely  true  method  of  measur- 
ing variations  in  the  exchange-value  of  money.  But  in  the 
world  as  it  is,  we  have  not  yet  reached  the  absolutely  true 
method. 

§  3.  We  can,  however,  be  sure  that  we  have  come  pretty 
near  to  it. 

In  the  first  place,  we  have  three  distinct  methods,  for  each  of 
which  much  can  be  said,  which  in  some  cases  regularly  give  the 
same  results,  and  which  in  all  ordinary  cases  give  results  very 
close  together.  The  cases  when  they  exactly  agree  are  when 
we  are  dealing  with  two  classes  equally  important  over  both 
the  periods. 

In  these  cases  we  may  first  prove  that  the  geometric  method 
(which  now  must  use  even  weighting)  exactly  agrees  with  the 
method  with  double  weighting  above  expounded.  The  con- 
dition to   be  observed  is  that  the  two  classes  are  such   that 


s/.Cj«ji/y/.,  =  "^ yji^y^f^^ ,  or,  which  is  the  same  thing,  x^o.^x/i.^  = 
yi^^yji^ .      From  tliis  condition  is  derived 

?/,/5i     •'•,«!     •Ti«,  -f  yA  ' 

Therefore  in  tiie  first  half  of  formula  (1)  for  the  method  with 
double  weighting,  applied  to  two  such  classes,  we  may  substitute 
one  of  tliese  values  of  that  half,  and  proceed  to  reduce  thus, 

y^2   ^X«i«2  +  yyAi%  =  M^ji2'^'W'i  +  ihy2^^A ) 

From  the  given  condition  is  also  obtained  x^x.^  =  y^y.^    '  ',  and 


COMPARISON    OF'    THE    THllEE    METHODS  403 

2/12/2  —  ^1^2  i^ii-     Therefore  the  last  expression  becomes 

«i  (2/12/2  ^  +  ^i2/2  -^AS)    «y/P;  (?/i2/2  ^|'^'  +  ^^?y.) 

Affain   from   the  o;iven  condition  is  obtained  >-^^=^-,  and 

S3      X  t)c 

-^-  =:^^.     Therefore  the  hist  expression  becomes 

«i«2      2/12/2 


/^2^«l«2  (^l2/2_+   ^l'«22/l2/2)  _  /I2        («1«2  _         I5        /i2 

«y/V2(^-'«i^22/i2/2  +  ^12/2)   '^i^M   "^^i  r^i' 

which  last  is  the  formnla  for  the  geometric  average  of  the  price 
variations  with  even  weighting.     Q.  E.  D. 

Next  we  may  prove  that  in  these  cases  Scrope's   emended 
method   likewise   exactly  agrees   witli   the  geometric  method. 

on  inserting  this  value  of  ^/x^x.^  in  the  formula  for  Scrope's 
emended  method  confined  to  the  two  classes,  and  reducing,  w*e 
get  

P2  ^  «2^^2/l2/2i'^l/^2  +  A^2/l2/2«l«2 

-^1  «y  2/12/2/^1^^2 + /5y2/i2/2«i«2 


s/a£^ ( ^o-A  +  ^'hf-^.^  _     k./l2      Q   E   D. 


Thus,  ^/•^^?i  tec  deal  with  two  c/r/.S'.sr.s-  equal/ 1/  iuiporfant  over  both 
the  periods,  the  method  with  double  iceighting,  and  Scrope's  emended 
method,  and  the  geometric  method,  all  yield  the  same  result. 

In  all  such  cases  (which  permit  both  the  money-sums  and 
the  mass-quantities  to  be  different  at  the  two  periods,  but  re- 
quire them  to  yield  equal  products)  the  three  methods  satisfy  all 
our  tests.  We  may  be  sure,  then,  that  in  these  restricted  cases 
the  common  result  is  the  true  one. 


404  THE    UXIVERSAL    METHOD 

§  4.  lu  other  ordiiuuy  cases  trial  shows  that  iu  their  results 
the  three  methods  do  not  diverge  considerably  from  one  another. 
Therefore  we  have  reason  to  bolie\'o  that  they  do  not  deviate 
considerably  from  the  truth. 

As  regards  their  divei'gence  amongst  themselves,  trial  seems 
to  sliow  that,  only  two  unequal  classes  being  employed,  the 
method  with  double  weighting  generally  gives  a  middling  re- 
sult. The  highest  is  given  by  the  geometric  method  and  the 
lowest  by  Scrope's  emended  method,  when  the  preponderating 
class  is  with  price  rising  above  the  general  average ;  and  re- 
versely the  lowest  by  the  geometric  method  and  the  highest  by 
Scrope's  emended  method,  when  the  preponderating  class  is 
with  price  falling  below  the  general  average.  With  more  classes 
this  rule  does  not  seem  to  hold,  unless  most  of  the  larger  ones 
have  prices  varying  in  the  one  direction  and  most  of  the  smaller 
ones  have  prices  varying  in  the  opposite  direction  (in  relation  to 
the  general  average  price  variation).  The  greatness  of  the  di- 
vergence is  determined  both  by  the  greatness  of  the  inequality 
in  the  sizes  of  the  classes  and  by  the  greatness  of  the  price 
variations — principally  by  the  latter.  It  is  greatest  when  the  pre- 
ponderating class  varies  little  and  the  smaller  class  varies  much  ; 
l^ut  it  may  also  be  considerable  when  the  preponderating  class 
varies  much  and  the  smaller  class  varies  little,  provided  the 
preponderance  is  not  excessive ;  for  when  it  is  excessive  all  the 
results  are  drawn  so  far  with  the  variation  of  the  excessively 
preponderating  class  that  their  divergence  may  be  lessened  al- 
most to  nothing, — and  such,  of  course,  is  the  event  also  in  cases 
where  the  classes,  in  pairs  or  sets,  are  nearly  equal  in  size.  In 
the  cases  of  moderate  inequality  in  the  sizes  of  the  classes  and 
of  excessive  variation  in  one  of  the  prices,  there  seems  to  be  a 
tendency  on  the  part  of  the  geometric  method  to  deviate  by 
itself,  becoming  untrustworthy,  while  the  other  two  methods 
keep  fairly  close  togetlier.  All  this  about  the  geometric  method 
in  general  agrees  with  what  has  already  been  found  to  be  the 
deviation  of  this  method,  iu  the  partial  cases,  in  compari- 
son   with    the    com[)()uent    parts  of    the  method    with   double 


COMPARISON    OF    THE    TIIIM^E    ^METHODS  405 

The  following  exann)lcs,  purposely  extrava<iaiit,  will  illustrate 
some  of  the  salient  positions. 

I  10  A  @  1.00     50  B  @  1.00  I  10  for  [A]     50  for  [B], 
II     6  A  @  1.50     70  B  %    .40  |     D  for  [A]     28  for  [B]. 

[B]    is  3.94  times  larger  than  [A].     The  geometric  method 

indicates  j^  =  0.5226  ;  the  method  with  double  weighting  makes 

^  1 
it  0.5240,  and  Scrope's  emended  method,  0.527-3. 

I  100  A  @  1.00     30  B  @  1.00  I  100  for  [A]     30  for  [B], 
II     80  A  @  1.50     40  B@    .20  I  120  for  [A]       8  for  [B]. 

[A]  is  7.071  times  larger  than  [B].     The  indications,  in  the 
same  order,  are  1.1G8G,  1.1547,  and  1.1370. 


I  100  A  @  1.00    3  B  @  1.00 
II    80  A  @  1.50    4  B  @    .20 


100  for  [A]     3       for[B], 
120  for  [A]     0.80  for  [B]. 


[A]    is    70.71    times   larger  than    [B].      The   indicjitions   are 
1.4585,  1.4555,  and  1.4515. 


I  100  A  @  1.00    300  B  @  1.00 
II     80  A  @  1.50    400  B@    .20 


100  for  [A]     300  for  [B], 
120  for  [A]       80  for  [B]. 


[B]    is    1.414  times  larger   than    [A].     The   indications   are 
0.4608,  0.4634,  and  0.4667. 


I  100  A  @    1.00    50  B  (aj,  1.00 
II     80  A  @  10.00     60  B@    .90 


100  for  [A]     50  for  [B], 
800  for  [A]     54  for  [B]. 


[A]    is    17.21    times  larger   than    [B] .     The   indications   are 
9.4630,  6.6824,  and  6.5369. 


I  100  A  @  1.00     100  B@1.00 
II     98  A  @  2.00     102  B@    .10 


100  for  [A]  100       for  [B], 
19Gfor[A]     10.20  for  [B]. 


[A]  is  4.38  times  larger  than  [B] .  This  example  may  be  com- 
pared with  one  in  Chapter  XI.  Section  IV.  §  3.  The  mass- 
quantities  have  varied  so  little  that  it  almost  comes  under  the 
cases  wdth  constant  mass-cpiantities.  If  it  did  so,  the  mass- 
quantities  of  the  first  period  remaining  constant,  we  know  with 

P.,  . 

certainty  that  the  price  variation  would  be  p^  =  1.05.    But  the 

variations  of  the  mass-quantities,  slight  as  they  are,  make  [B] 
larger  relatively  to  [A]  than  it  otherwise  would  be,  and  so  give 
greater  effect  to  the  fall  of  price.     Therefore  we  know  with  cer- 


406  THE  rxiVERSAi.  method 

taiiity  that  the  indication  .should  be  slightly  beh)\v  i.Oo.  Now 
the  method  with  double  weighting  yields  1.0442,  and  Scrope's 
emended  method  1.0405,  while  the  geometric  method  yields 
1.1404,  thus  being  certainly  wrong.  Which  of  the  former  two 
results  is  nearer  the  truth,  it  is  impossible  to  tell.  That  the 
geometric  method  should  fail  in  such  an  extravagant  example, 
ought  not  to  be  counted  much  to  its  discredit.  All  these  ex- 
amples being  extraordinary  in  their  variations,  the  general 
closeness  of  the  results  yielded  in  such  cases  by  the  three  methods 
is  a  warranty  of  their  greater  closeness  in  all  ordinary  cases. 

§  5.  In  the  second  place,  what  has  been  shown  of  the  geo- 
metric method  with  reference  to  the  partial  method  for  constant 
sums,  and  of  that  method  with  reference  to  the  truth,  evidently 
belongs  to  the  geometric  method  in  general,  and  to  the  method 
with  double  weighting,  which  contains  that  partial  method,  and 
also  to  Scrope's  emended  method,  which  likewise  comprehends 
that  method.  That  is,  there  is  neutralization  both  between  the 
many  classes  that  are  measured  together  and  between  the  suc- 
cessive periods  in  a  series.  In  consequence  of  this  last  quality, 
even  though  a  considerable  error  should  be  made  at  one  stage, 
there  is  probability  of  its  being  corrected  at  another,  and  there 
is  little  likelihood  of  the  error  being  any  greater  at  the  end  of 
a  long  series  than  near  the  beginning — except  in  case  of  con- 
tinual tendency  of  the  level  of  prices  in  one  direction,  although 
even  in  this  case  there  may  be  neutralization  through  changes 
in  the  sizes  of  the  classes.  A  complex  example  illustrative  of 
some  of  these  inductions  Avill  be  given  later. 

Lastly,  the  amount  of  the  errors  at  any  later  period  may  be 
subjected  to  a  certain  test,  wiiich  generally  shows  but  slight 
deviation — althougli  this  test  is  not  as  satisfactory  as  we  might 
desire.  This  test  is  to  suppose  the  period  in  question  to  be  fol- 
lowed by  a  period  with  everything  exactly  the  same  as  at  some 
earlier  period,  and  then  to  calculate  on  to  it,  to  see  how  the  re- 
sult serially  reached  for  it  compares  with  unity,  which  is  known 
to  be  the  true  result.  This,  however,  is  by  no  means  a  perfect 
test,  for  two  reasons.  The  one  is  that  the  error  proved  for  the 
supposed  period  is  not  necessarily  the  error  for  the  other  periods 


COMPARISON    OF    THE    TIIKEE    ISIETHODS  407 

whicli  preceded,  nor  a  definite  increase  upon  their  errors,  wlicuce 
their  errors  may  be  calculated.  For  there  is  no  gradual  ac- 
cumulation of  error,  but  irregularity,  and  some  of  the  preceding 
results  may  be  above,  and  some  below,  the  truth.  The  other 
reason  is  that  this  last  calculation  is  nothing  but  the  inverse  of 
a  direct  calculation  from  the  earlier  period  to  the  last  actual 
period ;  but  such  a  direct  calculation  we  know  to  have  no 
greater  validity  than  the  indirect  calculation.  Still,  the  fact 
that  in  practice  the  direct  and  the  indirect  comparisons  do  not 
diverge  much — and  especially  the  fact  that  they  do  not  diverge 
more  at  the  end  of  a  long  series  than  near  the  beginning,  is 
good  evidence  that  none  of  the  methods  deviate  much  from  the 
truth. 

§  6.  Between  the  three  methods  our  choice  may  be  guided  by 
what  we  have  so  far  learnt.  The  geometric  averaging  of  the 
price  variations,  with  single  weighting,  is  probably  the  least 
trustworthy,  because  we  have  seen  that  the  geometric  average 
between  more  than  two  equally  important  things  is  not  to  be 
depended  upon — and  we  have  sometimes  caught  it  flagrante 
delicto.  The  method  with  double  weighting,  using  the  geomet- 
rically measured  equivalent  mass-units,'' has  been  led  up  to  by 
a  chain  of  reasoning  which  seems  to  be  sound.  Yet  there  was 
something  defective  in  the  reasoning  at  an  early  stage,  since 
even  the  partial  method  for  the  cases  with  constant  sums  was 
found  to  fail  before  some  of  the  tests.  For  the  form  of 
Scrope's  method,  in  which  it  is  applied  to  the  geometric  means 
of  the  mass-quantities,  the  argument  is^that  this  method  under- 
lies both  the  partial  methods,  and  so  is  the  one  fundamental 
method,  which,  being  true  in  those  two^cases,  or  at  all  events  in 
one  of  them,  ought  to  be,  if  not  absolutely  true,  yet  near  to  the 
truth,  in  all  other  cases.  But  the  decisive  argument  for  it  is 
that,  while  it  stands  all  the  tests  that  are  satisfied  by  the 
method  with  double  weighting,  it  stands  still  another  test,  be- 
fore which  that  method  fails.  Hence  this'^method  (although 
the  geometric  method  shares  with  it  this  last  quality)  is,  in  all 
probability,  the  best  of  the  three. 

§  7.  The  fact  that  we  have  not  reached  a  perfectly  certain 


408  THE    UNIVERSAL    METHOD 

method — except  for  the  cases  when  tlie  mass-(iuantities  (the 
physical  parts  of  the  economic  world)  are  constant,  or  all  vary 
alike,  and  the  very  special  cases  when  there  are  only  two  eciually 
important  classes,  or  pairs  or  sets  of  snch  chisses,  not  to  mention 
the  cases  when  the  prices  are  constant,  or  all  vary  alike — must 
not  be  misinterpreted.  It  does  not  mean  that  in  cases  when  the 
mass-qnantities  and  the  prices  irregularly  vary,  there  is  no  one 
true  variation  of  money  in  general  exchange-value ;  for  if  that 
were  so,  there  could,  in  these;  most  common  cases,  be  no  varia- 
tion of  money  in  general  exchange-value  at  all,  which  is  absurd. 
What  it  means  is  that  our  mathematics,  so  far  as  yet  carried  in 
the  subject  of  averaging,  fail  us.  We  have  not  yet  found  the 
right  average — or  the  right  weighting  for  averages  already 
known.  It  may  be  there  is  no  average  that  is  perfectly  cor- 
rect— or  no  weighting  that  will  make  it  so.  Perhaps  no 
method  exists  to  be  found  that  will  stand  all  our  tests.  But 
from  the  fact  that  the  perfect  average,  and  the  perfect  weight- 
ing— or  the  perfect  method  of  combining  them — have  not  yet 
been  discovered,  it  does  not  necessarily  follow  that  they  are 
never  to  be  discovered.  Or  if  finally^  we  must  abandon  the 
search  as  hopeless  and  believe  that  no  perfect  method  exists  to 
be  discovered,  this  failure  of  mathematics  would  not  disprove 
the  existence  of  one  true  variation.  The  fact,  however,  that  we 
have  three  methods — not  to  mention  two  or  three  more,  as  will 
be  shown  presently, — which  in  all  ordinary  cases  give  results 
very  close  together,  and  which  we  have  every  reason  to  believe 
to  be  close  to  the  truth,  and  to  hold  the  truth  between  them, 
ought  to  make  us  fairly  content.' 

3  Here  miiy  be  inserted  a  suggestion  of  a  line  along  which  it  might  he  thought 
that  mathematicians  may  perhaps  be  able  finally  to  solve  our  problem  with  pre- 
cision, and  at  the  same  time  a  warning  against  over-expectancy.  When  the  mass- 
quantities  are  constant,  we  have  seen  that  the  solution  is  perfect.  Such  cases 
may,  then,  be  used  as  a  touchstone  for  the  rest.  Now  let  mathematicians  find 
the  weighting — according  to  means  of  some  sort  between  the  full  money-values  at 
each  of  the  two  periods  compared — which  will  make  the  geometric  average  of  the 
price  variations  always  agree  with  Scrope's  method  api)lied  to  the  constant  mass- 
quantities.  If  this  task  be  accomplished,  it  might  seem  as  if  the  geometric  aver- 
age of  the  price  variations  with  the  same  kind  of  weighting  would  universally  be 
correct,  including  the  cases  when  the  mass-quantities  vary.  An  approach  toward 
this  solution  maybe  indicated.  By  trial  it  is  found,  at  least  with  two  uneven 
classes,  that  the  geometric  average  of  the  price  variations  with  weighting  accord- 


OTHEE    METHODS    EXAMINED  409 


lY. 


§  1.  It  will  be  well  also  to  examine  other  methods,  but  es- 
pecially the  convenient  form  of  Scrope's  method  in  which  it  is 
applied  to  the  arithmetic  means  of  the  mass-quantities  at  each 
period,  or,  which  is  the  same  thing  still  more  conveniently,  to 
the  aggregates  of  the  mass-quantities  at  both  periods. 

Submitting  this  method  to  Professor  Westergaard's  test,  we 
have  for  the  indirect  comparison 

P3  ^  «2(^2  +  ^^2)  + /^2(2/i  +  ^2)  + 

Pj       a^{x^  +  a-,)  -\-  ^i^{y^  ^- y.^  +  ' 

«3(«^2  +  ^d  +  ['^■^2  +  2/3)  + 


«2(«'2  +  ^3)  +  ^'^2(^2  +  2/3)  + 


ing  to  the  arithmetic  means  of  the  full  money-values  at  each  period  gives  results 

with  error  opposite  to  the  error  given  by  the  geometric  average  with  geometric 

weighting.     For  example : 

100  A  @  1.00    100  B  @  1.00  I  100  for  [A]     100  for  [B]  -  200, 

100  A  @  1.90    100  B  @    .09  I  196  for  [A]        9  for  [B]  -  205. 

P,      205 
The  known  result  is  p^  =  ^,„  =  1.025.     The  geometric  weighting  is  4.666  for  [A] 

to  1  for  [B];  and  with  this  weighting  the  geometric  average  of  the  price  variations 
is  1.1426.  The  arithmetic  weighting  is  2.715  for  [A]  to  1  for  [B];  and  with  this 
weighting  the  geometric  average  of  the  price  variations  is  0.8552.  It  is  plain  that 
the  proper  weighting  must  be  given  by  some  mean  lying  between  the  geometric 
and  the  arithmetic.  One  such  mean  has  been  discovered  by  Gauss,  who  named 
it  the  arithmetico-geometric  mean.  If  the  classes  be  weighted  according  to  this 
mean,  the  weighting  will  be  3.657  for  [A]  to  1  for  [B];  and  with  this  weight- 
ing the  geometric  average  of  the  price  variations  is  1.014,  which  is  slightly  too 
low.  Other  examples  likewise  show  that  the  geometric  average  with  geometrico- 
arithmetic  weighting  comes  nearer  to  the  truth  than  the  geometric  average  with 
geometric  weighting ;  but  that  it  stili  errs  on  the  side  of  the  geometric  average 
with  arithmetic  weighting.  In  our  examjile  the  proper  weighting  for  the  geo- 
metric average — that  which  will  make  it  give  the  true  answer — is  3.7526.  A  still 
closer  approach  to  this  is  made  by  a  mean  which  is  a  combination  of  all  the  three 
common  means,  and  wliich  may  be  called  the  arithmetico-geometrico-harmonic 
mean.    This  is  the  harmonic  mean  between  the  ai'ithmetic  and  geometric  means 

,     ^        ,  .  ,    .     2{a+hy'ab 

between  the  two  quantities  in  question,  the  formula  for  which  is 7=  . 

a  +  6  +  2»'^a6 

Weighting  the  classes  according  to  this  mean  between  their  two  full  money-values, 
we  find  the  weight  for  [A]  to  be  3.718  times  that  for  [B].  With  this  weighting 
the  geometric  average  of  the  price  variations  is  1.0201.  In  practice,  however, 
this  method  would  hardly  differ  from  the  preceding,  which  would  differ  very 
slightly  from  the  completely  geometric  method.  There  is,  moreover,  a  consider- 
ation which  invalidates  the  idea  that  a  mean  perfectly  good  for  our  purpose  exists  : 
for  if  the  money-values  of  all  the  classes  be  constant,  all  kinds  of  means  are  the 
same,  and  all  are  then  defective,  as  proved  in  the  next  preceding  Chapter  on  the 
method  for  constant  sums. 


410  TIIH    UNIVERSAL    METHOD 

while  the  direct  eoniparisoii  is 


Pj      «j(ic^  +  x^  +  A(yi  +  2/3)  + ' 

These  regularly  agree  :  (1)  if  o..^  =  a.^  =  «j,  and  so  on,  whatever 
be  the  variations  in  the  mass-quantities  ;  (2)  if  x^  =  x^  =  x^,  and 
so  ou,  whatever  be  the  variations  in  the  prices ;  (3)  if  a^  =  a.^, 
provided  x^  =  x.^,  and  so  on,  or  if  a.,  =  a^,  provided  x^  =  x^,  and 
so  on  ;  (4)  if  a^  =  «„  provided  x^  =  x^,  and  so  on.  These  are 
the  four  divisions  in  which  Scrope's  method  apjilied  to  the  geo- 
metric means  and  the  method  with  double  weighting  regularly 
agree ;  but  in  each  division  only  half  of  the  conditions  are  re- 
tained, uniform  variations  in  the  prices  or  in  the  mass-quan- 
tities being  cut  off.  In  a  series  of  four  or  more  periods  it  is 
easily  seen  that  there  will  be  necessary  agreement  only  (1)  if 
there  are  no  variations  in  the  prices,  or  (2)  if  there  are  no  varia- 
tions in  the  mass-quantities,  or  (3)  if  the  periods  fall  into  two 
sets  in  each  of  which  there  are  no  variations  at  all.  In  short, 
this  method  behaves  in  this  matter  exactly  like  Professor  Lehr's 
method. 

§  2.  Still  other  limitations  may  be  discovered  in  this  method, — 
by  which  limitations  also  Professor  Lehr's  method  will  be  found 
to  be  restricted.  The  investigation  may  be  opened  with  examina- 
tion as  to  what  is  the  relation  between  the  arithmetic  means,  or 
between  the  aggregates,  of  the  mass-quantities  of  the  different 
classes  in  cases  when  the  result  is  known  with  certainty.  These 
are  when  two  classes  are  dealt  with  tiiat  are  equally  important 
over  both  ])eriods.  For  simplicity  we  shall  begin  with  cases 
where  the  result  is  unity,  indicating  constancy  in  the  exchange- 
value  of  m(mey ;  and  always  we  shall  follow  the  usual  ])ractice 
of  employing  mass-units  that  are  equivalent  at  the  first  period. 
The  following  is  the  simplest  schema  : 

I  100  A  @  1.00     200  B  @  1.00  I  100  for  [A]     200  for  [15]  —  800, 
II  100  A  @  2.00     200  B  @    .50  |  200  for  [A]     100  for  [B]  —  300. 

200  400 

Here  the  mean,  or  aggregate,  mass-quantities  purchased  over 
both  the  periods  together  are  twice  as  many  of  these  mass-units 


OTHER    METHODS    EXAMINED 


411 


of  [B]  as  of  [A] .      Tliis  is  in  accordance  with  tlie  formula  dis- 
covered iu  Chapter  IX.  Section  II.  §  5,  namely, 


,       ^'is'  -  1) 


;/  = 


for  here 


^->%      ' 


,       -^'(^  -  1) 


y  = 


i-i 


=  2.t' 


Now  the  same  result,  constancy,  aj)pears  also  if  things  hap- 
pened as  represented  in  the  following  schemata : 

I  100  A  @  1.00     100  B  @  1.00     100  for  [A]     100  for  [B]  —  200, 


and 


II     50  A  @  2.00  200  B  @    .50 

150  300 

I  100  A  @  1.00  150  B  @  1.00 

II     75  A@  2.00  200  B@    .50 


and 


175 


350 


I  100  A  @  1.00     300  B  @  1.00 
II  150  A  @  2.00     200  B  @    .50 


100  for  [A]     100  for  [B]  —  200; 


100  for  [A]     150  for  [B]  —  250, 
150  for.[A]     100  for  [B]  —  250; 


100  for  [  A]     300  for  [ B]  —  400, 
300  for  [A]     100  for  [B]  —  400; 


250 


500 


and  an  indefinite  number  more  of  such  examples  might  be  made. 
In  all  these  it  is  observable  that,  while  the  states  of  things  per- 
mit of  even  weighting,  and  the  price  variations  are  geometric, 
wherefore  the  result  is  known  to  be  constancy,  with  the  further 
condition  that  the  total  sums  of  money  spent  on  both  the  classes 
together  at  each  period  are  the  same,  the  aggregates,  and  hence 
the  arithmetic  means,  of  the  mass-quantities  of  each  class  pur- 
chased over  both  the  periods  together  are  twice  as  many  of  [B] 
as  of  [A] .  This  constant  relation  of  two  A's  to  one  B  (these 
being  equivalent  at  the  first  period)  is  comparable  with  the  rise 
of  A  in  price  from  one  to  two  money-units.  Now  under  these 
conditions  this  relation  between  the  arithmetic  means,  or  ag- 
gregates, of  the  mass-quantities  of  each  class  over  both  the 
periods  may  be  proved  to  be  universal.     The  general  schema  is 

It/A@1         y'B(^l      I       Vfor[A]  y/ for  [B]  —  x/ +  ?//, 

II  x/  A  @  «/     2//  B  @  /?/  I  x./a./  for  [A]     y//3/  for  [B]  -  x/a/  -f  y//?/; 


412  THE    UNIVERSAL    METHOD 

iu  which   tlie  coiulitiuiis,  beside   the  equivalence  of  the  mass- 
units  at  the  first  period,  are  :  (1)  x^'x.^o..^  =  y^'y^^.^' >  (^)  N 

=     , ,  (3)  x./a./  +  ?//,5/  =  x/  +  ///.      We  wish  to  prove  that 

2 

From  the  first  and  second  conditions  is  derived 


*  ry»   /y*  rg*    '  rv%    '  ft    '  ^  Hf*       O" 


and  the  third  condition  reduces  thus, 


^2'  =  ^'/s'- 


x'x'  o.„ 


From  the  first  condition  is  derived  \i'  =    ^  ,\  ?'-  ;  wherefore 

^'       y-1^2 


-'1   ^2  '^2  "^2  /       / 

"1  nn  I  n   '  C-*  I  /?   '  2        2    • 


1        2       2  "^ 

<«2^''^/C' 


Therefore  the  sums  of  these  equals  are  equal.  Q.  E.  D.  This 
shows,  incidentally,  that  our  ])revious  formula,  which  with  the 
relation  between  the  prices  above  sup])osed  in  the  second  condi- 
tion reduces  as  follows, 

is  true,  not  only  for  the  ])urp()se  for  which  it  was  invented,  but 
also,  witli  y'  =  y^'  -\-  y./  and  x'  =  x/  +  y./ ,  for  the  i)urpose 
in  Avliich  Ave  are  at  present  interested.  Thus  in  our  particular 
examples,  where  a,^'  =  2.00,  the  arithmetic  mean  number  of  B's 


OTHER    METHODS    EXAMINED  413 

purcliasetl  over  both  the  periods  will  always  be  twice  the  mean 
number  of  A's,  when  these  conditions  are  fulfilled.  And  on 
these  relative  mass-quantities  Scrope's  method  always  indicates 
constancy  ;  for 

P,  _  1x2+2x1  _  2^1  _  .  . . 
P,       1x1  +  2x1       l  +  2~-' 

And  so  in  the  universal  cases  :  applied  to  the  universal  rela- 
tions between  the  mass-quantities,  when  the  above  conditions 
are  fulfilled,  Scrope's  method  always  indicates  constancy,  and 
therefore  always  yields  the  right  result ;  for 


P^       1  X  «/  +  a/  X  /9/       «/  +  1 


=  1.00. 


P^  ;     1  X  1  +  «/  X  1        1  +  «/ 

Now,  if  we  used  other  mass-units,  we  should  get  different  re- 
lations between  their  aggregate  numbers  in  the  two  classes,  but 
the  new  aggregate  numbers  would  be  equally  constant.  In 
general  we  know  that  the  results  obtained  by  Scrope's  method 
are  the  same  whatever  be  the  mass-units  used.  Therefore, 
what  has  above  been  proved  of  Scrope's  method  using  certain 
mass-units  may  be  universalized  of  it  using  any  mass-units,  and 
we  have  this  general  proposition  :  When  the  conditions  are  such 
that,  with  two  classes,  their  prices  varying  from  unity  to  the  op- 
posite geometric  extremes,  the  weighting  is  even  over  both  the  pe- 
riods, wherefore  we  know  that  the  geometric  mean  of  the  price 
variations  is  true,  and  it  here  indicates  constancy,  then,  provided 
further  that  the  total  sums  spent  on  the  two  classes  together  at  each 
period  are  the  same,  also  Scrope's  method  applied  to  the  arithmetic 
mea)is  (or  to  the  aggregates)  of  the  mass-units  of  each  class  pur- 
chased at  each  period  is  always  correct.  It  will  be  noticed  that 
the  cases  are  extremely  limited  in  which  Scrope's  method,  so  used, 
can  be  exactly  right,  except  by  chance. 

§  3.  Let  us  now  widen  the  restriction  by  leaving  off  the  last 
condition.  We  may  take  another  numerical  example,  with  the 
same  price  variations  as  in  the  preceding,  but  with  different 
mass-quantities.  Things  might  happen  as  represented  in  this 
schema  : 


414 


THE    UNIVERSAL    .METHOD 


I  100  A  O3  1.00       SOBCr/   1.00 
II     88  A  On  2.00     440  B  Qv,    .50 

188  520 


100  for  [A]       80  for  [B]  — 180, 
176  for  [A]     220  for  [K]  —  396. 


Here  Ave  likewise  have  even  weighting  over  both  the  periods 
(as  100  X  176  =  88  x  220),  so  that  we  know  that  the  geo- 
metric mean  of  the  price  variations  is  correct,  and  this  still  indi- 
cates constancy, — as  is  also  indicated  by  Scrope's  method  itself, 
applied  to  the  geometric  means  of  the  mass-(piantities.  But  the 
arithmetic  means  (or  the  aggregates)  of  the  mass-quantities  of 
each  class  over  both  the  periods  are  no  longer  in  the  relation  of 
2  B's  to  1  A.  And  now  Scrope's  method,  applied  to  the  arith- 
metic means  of  these  mass-(|uantities,  thus 


94  X  2  +  260  X  i 


;18 


94  X  1  +  260  X  1       354 


=  0.8983, 


indicates  a  fall  of  10.17  per  cent.,  which  is  far  astray. 

But  the  above  numerical  schema  may  be  analyzed  into  the  fol- 
lowing: : 


I  100  A  @  1.00  80  B  @,  1.00 

■40  A  @  2.00  200  B@    .50 

IlU0A@2.00  200  B@    .50 

8  A  @  2.00  40  B  @    ..50 


100  for  [A]  80for  [B]  — 180, 

80  for  [A]  100  for  [B]  —  180 ] 

80  for  [A]  lOOfor  [B]  — 180| 

IGfor  [A]  20  for  [B]  —    3gJ 


Here  we  see  that  at  the  second  period  after  we  have  spent  the 
same  total  sum  of  money  as  at  the  first — 180  money -units  (but 
in  money,  notice,  that  is  shown  to  have  the  same  exchange- 
value  at  both  the  periods), — the  fact  that  we  go  on  si)en(ling 
more  money  on  the  two  classes  in  the  same  proportions  in  no 
wise  affects  the  exchange-value  of  our  money  at  this  period. 
Hence  all  that  we  are  concerned  with  in  the  cora})arison  is  ex- 
pressed in  the  following  : 


I  100  A  @  1.00       80  B  @  1.00 
II     40  A  @  2.00     200  B  @    .50 


100  for  [A] 
80  for  [A] 


80  for  [B]— 180, 
100  for  [B]— 180, 


140 


280 


And  here  we  again  have  the  same  relations  that  we  had  in  our 
earlier  examples,  the  arithmetic  mean  (or  aggregate)  number  of 
B's  purchased  over  both  the  periods  being  twice  that  of  the  A's. 


OTHER    METHODS    EXAMINED  415 

Therefore  Scrope's  method  a})pliecl  to  the  arithmetic  means  (or  to 
the  aggregates)  of  these  numbers  of  mass-units  gives  the  true  re- 
sults— and  oidy  when  applied  to  these. 

That  Scro])e's  method,  so  applied,  is  always  correct  when 
these  conditions  are  fulfilled,  may  also  be  demonstrated.  The 
universal  schema  is  the  one  above  used  ;  but  now  the  conditions 
are  only   the  former  two,   namely  (1)  x^'x.^a.^  =  y^'y.! ["^^  ^  '^"^ 

(2)  [ij  =   -^.     Now  if  we  reduce  the  total  sum  spent  on  both 

the  classes  at  the  second  period,  namely  x,^o..,'  -f  y.li'i<l  -,  to  what 
is  spent  on  them  at  the  first  period,  namely  x^  -f-  y.^ ,  we  may 
do  so  by  reducing  the  particular  sums  spent  on  [A]  and  on  [B] 
at  the  second  period  in  the  same  proportion,  and  consequently 
also  the  numbers  of  the  mass-units  purchased  of  [A]  and  of 
[B]  at  the  second  period  likewise  in  the  same  proportion  ;  and 
doing  this,  we  get  the  following  schema  : 

I  a;/  A  @  1  2//  B  @  1 

I  V  for  [A]  2//  for  [B], 

From  the  given  conditions  we  derive  x^x.^a.^  =  y^'y^^     /  ^^^ 


1 


I  Q  /2 
2 


x^'x.^ap  =  y^'yj ,  and  again  y^y.^^-i^  =  x^'x./  ^  ,  and  y^'yj [i 

/  2 
^  x^'x^  ;  and  by  reducing  the  expressions  for  the  sums  spent  on 

[A]  and  [B]  respectively  at  the  second  period,  and  by  substi- 
tuting these  values,  the  former  reduces  to  y^' ,  and  the  latter  to 
x^'  ;   wherefore  simpler  expressions  for  the  numbers  of  A's  and 

y' 

B's  purchased  at  the  second  period  are  - ,  or  y^' fi.,'  for  the  A's 
and  -^  or  x'aJ  for  the  B's,  and  the  schema  becomes 

/^2 


I  x/  A  @  1  ?//  B  @  1 

II  ^  A  @  «./      r/«/  B  @  /32 


a:/  for  [A]     ?//  for  [B], 
2// for  [A]     Vfor[B]; 


and  the  total  of  tlie  numbers  of  A's  purchased  over  both  the 


416  THE    UNIVERSAL    METHOD 

X  '  o.  '  -\-  u  ' 
periods  is     '    -    -. — '-^ ,  aud    that  of  the    numbers    of  B's  is 

ic/ttg'  +  ?y/>  •'allowing  that  tlie  B's  are  a^  times  the  A's.  Now 
Serope's  method  applied  to  these  relative  total  mass-quantities, 
and  therefore  also  if  applied  to  the  arithmetic  means,  always  in- 
dicates the  right  answer,  constancy,  as  above  shown.     Q.  E.  D. 

Thus  we  find  that  in  all  cases  where,  with  two  classes  equally 
important  over  both  the  periods,  the  exchange-value  of  money 
remains  constant,  a  condition  for  Serope's  method  applied  to 
the  arithmetic  mean  of  the  mass-quantities  being  correct  is  that 
we  must  make  use  only  of  those  mass-quantities  which  are  to- 
gether purchased  at  each  period  with  the  same  total  sum  of 
money/  Hence  we  may  induct  that  in  all  cases,  when  the 
exchange-value  of  money  remains  constant,  a  condition  of 
Serope's  method,  so  applied,  bemg  correct,  is  that  we  must 
apply  it  only  to  the  mass-quantities  which  we  have  so  reduced. 
Thus  a  necessary  preliminary  loorJc  for  the  employment,  with  ex- 
pectaiimi  of  the  best  results,  of  Sorope\s  method  applied  to  the 
arithmetic  means  of  the  mass-quantities,  is  that  toe  must  reduce  the 
mass-quantities  to  those  which  are  purchased  at  both  periods  ivith 
the  same  total  sum  of  money  (^provided  this  be  constant  in  ex- 
change-value). 

Of  this  another  brief  proof  may  be  supplied  by  the  use  of  a 
test  case.     Suppose  these  states  of  things  : 


I  100  A  @  1.00     100  B  @  1.00 

II    75  A  @  1.50     130  B@    .GO 

III    50  A  @  1.00      50  B@  1.00 


100       for  [A]     100  for  [B]  —  200. 
112.50  for  [A]       78  for  [B]  —190.50, 
50       for  [A]       50  for  [B]  —  100. 


Here  it  is  evident  that  money  has  the  same  exchange-value  at 
the  third  as  at  the  first  period,  and  this  ought  to  be  indicated  by 

^  The  same  reduction  is  required,  in  these  cases,  also  if  we  should  use  Serope's 
method  applied  to  the  smaller  mass-quantities  at  either  period,  that  is,  to  the 
mass-quantities  common  to  both  the  periods.  Tlien,  under  the  conditions  above 
supposed,  this  method  likewise  gives  the  right  results  ;  for,  supposing  «,'  >  1  and 

So'  <  1,  the  smaller  mass-quantities  are  —.  A  aud  2/i'B,  that  is,  aJ  times  more  B's 

than  A's,  as  before. — Also,  in  the  same  cases,  the  geometric  average  of  the  price 
variations  with  the  weighting  of  the  smaller  total  exchange-value  of  each  class  at 
either  period  will  always  give  the  riglit  result ;  for  the  smaller  total  money-values 
are  y^  for  each  class,  that  is,  the  same,  so  that  even  weighting  will  have  to  be 
used.    Thus  the  same  reduction  is  required  here  also. 


OTIIKi;    MI'/niODS    EXAMINED  417 

the  two  results  obtained  for  the  two  sets  of  priee  variations.  ( )ur 
three  su})eri()r  methods  give  tlie  correct  final   answer;  for  the 

P  P 

geometric  method  indicates    /  =  ().!>!)( )2  and  ^  =  1.0098,  the 

'    1  2 

P  P 

method  with  double  weightinj*-     '=  0.9800  and   ^^  =  1.0111, 

1  2 

P  P. 

and  Scrope's  emended  method  p^  =  0.9885  and  p'*=  1.0116, 

P 

whence  in  every  instance  p^  =  1.00.      Not  so  Scrope's  method 

^  1 
applied  to  the  arithmetic  means  of  the  mass-quantities;  for  its 

P  P  P 

results  are  ./  =  0.9888  and    /  =  1. 0;}21, whence   /=  1.0206, 

^1  ^1  ^1 

wrongly  indicating  a  rise  of  2.06   per   cent.^     But   if  all   the 

mass-quantities  are  reduced  to  those  which,  in  the  same  propor- 
tions as  actually  purchased  in,  were  purchasable  at  every  period 
with  the  same  total  sum  of  money,  applied  to  their  arithmetic 

P.,  P 

means  Scrope's  method  yields  p^' =  0.9922  and  p^  =  1.0078, 

P 

whence  p^=  1.00.^ 

This  exanqjle  is  one  in  which  Professor  Westergaard's  test  has 
been  shown  to  be  satisfied  by  our  three  methods  (all  containing  a 
geometric  element),  but  not  by  any  of  the  purely  arithmetic 
methods — Professor  Lehr's  included, — illustrating  condition  (4) 
above  in  §  5  of  Section  II.  and  at  the  commencement  of  this  Sec- 
tion. It  is  plain,  then,  that  also  Professor  Lehr's  method  requires 
a  similar  reduction  of  the  mass-quantities  before  it  can  bring 
out  its  best  results.  In  the  examj)le  just  given  this  method  em- 
ployed in  the  direct  comparison  of  the  third  with  the  first  period 
shows  sameness  of  exchange-value  of  money  at  the  third  as  at 
the  first.     But  employed  in  measuring  each  variation  separately, 

-  Applied  to  the  smaller  mass-quantities  at  either  period  in  eanh  comparison, 
Scrope's  method  is  still  more  wrong  in  these  cases ;  for  then  its  indications  are 

P  P  P 

5"^  =  0.0857,  :p-^  =  0.9523,  and  ^  =  0.9387. 

JTi  1  2  "l 

3  Applied,  as  before,  to  the  smaller  of  these  reduced  mass-quantities  at  either 
period,  Scrope's  method  again  gives  the  right  final  result ;  for  its  intermediate 

P  P 

indications  now  are  p^  ==  0.9964  and  p-^  =  1.0()35. 

27 


418  THE    rMVKRSAL    MKTMOD 

P  P  P 

its    results    are      -  =  0.9SS0    and       '' =  ().l»i)  1  C,    whence   ./ = 

0.9797,  wrongly  indicating  a  fall  of  2.()'>  per  cent.  Yet  ap- 
plied to  the  niass-(|uantities  reduced  as  l)efore  at  every  period,  it 

1'. 
is  consistent  in  always  yielding  .,'=  1.01)  ;   for    its  two    serial 

1 

J>.  P. 

indications  now  are  reciprocals,  being      -  =  ().!IS9.")   and  p' = 

1.0106.  Also  in  the  several  examples  al)ove  given  (in  §  2), 
which  are  variations  upon  one  theme,  in  all  those  in  which  the 
total  sums  at  each  period  are  the  same,  Professor  Ijchr's  method 
rightly  indicates  constancy  ;  but  in  the  one  in  which  the  total 
sum  spent  at  the  second  period   is  larger  tiian  at  the  first,  Pro- 

P., 

fessor  Lehr's  method  yields  p"  =  1.1  OS  1,    which    is   10.81    per 

cent,  too  high.^  Thus  in  order  to  get  the  result  known  to  be 
true,  Professor  Lehr's  method  requires  a  preceding  reduction 
(except  in  rare  cases  when  the  })roper  conditions  happen  to  ex- 
ist) before  it  is  to  be  applied — a  preliminary  labor  the  need  of 
which  its  author  never  contemplated. 

§  4.  When  money  has  not  the  same  exchange-value  at  the 
second  period  as  at  the  first,  one  of  the  conditions  necessary  for 
proving  the  correctness  of  Scrope's  method  used  in  the  way  under 
discussion  (and  also  of  Professor  Lehr's  method)  is  absent — since 
constancy  of  money's  extihange-value  resulted  from  the  two 
conditions  posited  in  the  above  demonstration  and  was  indispen- 
sable to  it.  Hence  even  if  we  make  this  preliminary  reduction, 
— or  if  it  is  superfluous,  the  state  of  things  already  being  as  de- 
sired,— we  (iaimot  use  the  above  demonstration  to  justify  a  belief 
that  in  all  cases  Scrope's  method  applied  to  the  arithmetic 
means  of  the  mass-(piantities  will  give  the  correct  answer.  A 
single  negative  instance  is  sufficient  to  disjicl  any  such  belief. 
We  may  suppose  the  following  state  of  things  : 


I  100  A  @  1.00    100    B@1.00 
II     50  A  @,  2.00     133JB@,    .75 


100  for  [A]     100  for  [B]  —  200, 
1 00  for  [ A  ]     1 00  for  [  B]  —  200. 


*  Cf.  this  result  with  that  ahove  given  for  Scrope's  method  applied  to  the  arith- 
metic means.  From  these  and  some  other  examph^s  it  would  seem  as  if  this  form 
of  Scrope's  mctkixl  and  Le/ir'.i  infihod  err  nearly  equitlly  on  opposite  sides  of  the 
truth. 


OTI[Eri    MKTIIons    EXAMiXKl)  419 

As  the  weighting   is  even  and   only  two  classes  arc  dealt  with, 

this  is  a  case  in  which  all  the   three   snperior   methods  agree. 

Their  common  resnlt  is  most   easily  (il)tain('d   hy  the  geometric 

P, 
method.     This  is  ,/  =  \^'l  x   'l  =  ^^  =  1.2247,  indicating  a 

rise  by  22.47   per  cent.      But  Scrope's  method  ai)]»licd   to  the 
arithmetic  means  of  these  mass-quantities  is 


P^  ^  75  X  2  +  llOf  X  I  ^  237^ 
P  ~75  X  1  4-  n<)S  X  1       19 If 


=  1.2;5!n 


which,  indicating  a  rise   by   23.91    per   cent.,  is   slightly  above 
the  truth. ^ 

For  further  comparison  another  more  irregular  example  may 
be  subjoined  : 


I  40  A  @,  1.25       70B^j,  1.50 
II  50  A  (^y  1.80     100  B@  1.30 


50  for  [A]     105  for  [B]  —  155, 
90  for  [A]     130  for  [B]  —  220. 


Here  the  geometric  method  (in  which  the  weighting  is  1  for  [A] 
and  1.7416  for  [B])  indicates  a  rise  of  the  price-level  by  4.30 
per  cent.,  the  method  with  double  weighting  a  rise  by  4.31 
per  cent.,  and  Scrope's  emended  method  a  rise  by  4.33  per  cent. 
— ^all  three  almost  exactly  alike.  On  the  arithmetic  means  of 
the  full  mass-quantities,  Scrope's  method  indicates  a  rise  by  4.21 
per  cent.;  and  on  the  arithmetic  means  of  the  mass-quantities 
reduced  so  that  their  total  money-values  are  the  same  at  both 
periods  (the  numbers  of  A's  at  the  second  period  being  reduced 
to  35.225,  and  that  of  B's  to  70.45)  Scrope's  method  is  still 
slightly  wrong,  as  it  indicates  a  rise  by  4.36  per  cent.'' 

AVhat  is  needed  more  thau  a  reduction  of  the  total  money- 
values  to  the  same  figure  is  the  reduction  to  the  same  figure  of 
the  total  exchange-value.^.  Thus  the  first  of  these  exam])les 
would  need  to  be  altered  into  the  following  : 

5  Applied  here  to  the  smaller  mass-quantities  at  either  period,  Scrope's  method 
indicates  a  rise  by  only  lOs  per  cent. — a  considerable  error. — Lehr's  method  here 
indicates  a  rise  by  21.05  per  cent. 

*  Applied  here  to  the  smaller  of  the  original  mass-quantities  at  either  period 
Scrope's  method  indicates  a  rise  by  r).l(j  per  cent.;  and  applied  to  the  smaller  of 
the  reduced  mass-quantities,  a  rise  by  3.60  per  cent. — Lehr's  metliod  applied  to 
the  original  mass-quantities  indicates  a  rise  by  4.4(j  per  cent.;  and  applied  to  the 
reduced  mass-quantities,  a  rise  by  4..'{.S  per  cent. 


420  THE  univp:rsai.  method 

I  100         A  (w,  1.00     100         B  @  1.00 
II     61.235  A  @  2.00     163.293  B  @     .75 

1100        for  [A]     100        for  [B]— 200, 
I  122.47  for  [A]     122.47  for  [B]  — 244.94. 

And   now   Scrope's   method   ou   the  arithmetic  means  of  these 
mass-quantities  is  : 

P,      80.()17  X  2  -I-  131.646  x  | 


80.617  X  1  +  131.646  x  1 


=  1.2247 


which  is  exactly  right.'  This  exact  agreement,  however,  is  due 
to  some  peculiarity  in  this  example  ;  for  we  do  not  find  it  al- 
ways. Thus  in  the  second  example,  after  reducing  the  total 
sum  at  the  second  period  to  161.68  (the  number  of  A's  then  be- 
ing 36.745  and  that  of  B's  73.49),  on  the  arithmetic  means  of 
the  mass-(|uantities  Scrope's  method  indicates  a  rise  by  4.27  per 
cent.*  But  we  have  reason  to  believe  that  this  method,  so  used, 
would  generally  give  a  more  nearly  correct  result  than  when 
applied  to  the  arithmetic  means  either  of  the  full  mass-quantities 
as  they  happen  to  be  at  each  period  or  to  the  mass-cpiantities 
reduced  merely  to  the  same  total  money-values.  Still,  to  em- 
ploy Scrope's  method  in  this  way  recpiires  that  we  should  first 
know  the  constancy  or  variation  of  the  exchange-value  of  money. 
Hence  Scrope's  method,  so  used,  is  unsuitable  for  finding 
the  c(mstancy  or  variation  of  the  exchange-value  of  money, 
since  this  use, of  it  presupposes  the  knowledge  we  are  .search- 
ing for. 

Nor  can  the  method  of  apj)roach  be  employed  with  any  great 
security.  Thus  in  the  first  of  the  above  examples,  after  the 
measurement  by  Scrope's  method  a])])licd  to  the  arithmetic  mean 
of  the  mass-quantities  as  they  actually  are,  which  has  been  seen 
to  indicate  a  rise  by  2.'>.!*1  ])er  cent.,  if  we  provisionally  assnme 
that  prices  have  so  risen,  we  might  work  over  the  schema  into 
the  following  : 

'  Here  also  the  right  result  is  given  both  by  Scrope's  method  applied  to  the 
smaller  of  these  mass-quantities  at  either  period,  and  by  Lehr's  method. 

*This  calculation  has  been  made  on  the  assumption  tliat  the  indication  given 
by  the  method  witli  double  weighting,  at  1.04.31,  was  the  right  one. — On  these 
reduced  mass-quantities  Lehr's  method  now  indicates  a  rise  by  4.34  per  cent. 


OTPIER    METHODS    EXAMINED  421 

I   100       AT";)  1.00     100         BOh  1.00 
II     Ol.Oo  A@,2.00     165.213  Br^    .75 

1100       for  [A]     100       for  [H]  — 200, 
I  123.91  for  [A]     123.91  for  [B]  —247.82; 

and  now,  applied  to  the  arithmetic  means  of  these  mass-quan- 
tities, Scrope's  method  indicates  a  rise  by  20.04  per  cent. 
Again  assuming  this  to  indicate  the  price  variation,  we  might 
repeat  the  operation  thus  : 

I  100       A  @  1.00     100         B  ®  1.00 
II     (50.02  A  @  2.00     160.053  B@    .75 

100       for  [A]  100       for  [B]  — 200, 

120.04  for  [A]  120.04  for  [B]  —240. 08; 

whence  the  indication  is  of  a  rise  by  22.61  per  cent.  And 
again  : 

I  100         A  @  1.00    100       B  @  1.00 
II     61.305  A  @  2.00    163.48  B  @    .75 

100       for  [A]     100       for  [B]— 200, 
122.61  for  [A]     122.61  for  [B]  —245.22; 

whence  tiie  indication  is  that  prices  in  general  have  risen  by 
22.4(5  per  cent.  And  if  we  repeated  the  operation  once  more, 
M^e  should  practically  get  the  same  result  again,  or  perhaps  a 
closer  approa(;h  to  22.47,  according  to  the  extent  to  which  we 
carry  the  decimals.  Therefore,  having  got  practically  the  same 
result  from  two  successive  calculatiou.s,  we  should  have  no 
reason  to  go  further,  and  should  adopt  this  result, — which  we 
already  know  to  be  the  right  one.  But  here  again  there  is 
something  peculiar  in  this  example,  that  permits  the  getting 
of  the  right  result ;  for  the  right  result  is  not  always  got  in 
this  way.  Thus  in  the  second  of  the  above  examples,  for 
which  the  first  employment  of  Scrope's  method  applied  to  the 
arithmetic  means  of  the  full  actual  mass-quantities  gave  us  a 
rise  by  4.21  per  cent.,  if  we  assume  this  to  be  right  and  reduce 
the  total  money-values  accordingly  (that  of  the  second  period  to 
101.5255),  and  work  out  as  before,  the  indication  is  of  a  ri.se  by 
4.34  per  cent.;  and  again,  assuming  this,  and  reducing  the  total 
money- values  (of  the  second  period  to  161.727),  and  working 
out  as  before,  the  indication  is  of  a  rise  by  4.53  per  cent.';   and 


422  THE    UNIVKIJSAI.     M  F.'I'IK  H  ) 

next,  after  reducing  the  total  money-values  (of  the  second 
j)eriod  to  102.0215),  the  indication  is  of  a  rise  by  4.35  per 
cent.  This  fourth  answer  hardly  differs  from  the  second, 
wherefore  the  next  would  hardly  diifer  from  the  third,  and 
pursuing  these  operations  indefinitely  we  might  perhaps  reach 
some  unvarying  result  somewhere  between  4..'j4  and  4.53,  but 
never  could  we  get  the  right  result,  which  lies  outside  these 
limit«.  Yet  even  if  the  right  result  could  be  reached  in  this 
way,  the  method  would  be  unserviceable,  as  no  more  laborious 
method  can  be  imagined. 

§  5.  The  peculiarity  in  the  first  example  which  caused  it 
twice  to  yield  the  happy  result,  whil(>  the  results  in  the  other 
example  were  always  divergent,  is,  in  all  probability,  the  same 
peculiarity  in  it  that  caused  the  three  superior  methods  exactly 
to  agree.  This  is  the  fact  that  in  it  the  t^vo  classes  are  equally 
important  over  both  the  i)criods.  Thus,  in  general,  when  two 
classes  equally  important  over  both  the  periods  are  dealt  with, 
the  same  result  is  given  not  only  by  the  superior  methods,  but 
also  by  Scrope's  method  applied  to  the  arithmetic  means  of  the 
mass-quantities  after  these  have  been  reduced  so  as  to  be  pur- 
chased at  each  period  with  sums  of  money  possessing  the  same 
exchange-value,  and  by  Professor  Lehr's  method  a]>plied  to  the 
mass-quantities  similarly  reduced,  and  even  by  the  former  of 
these  methods  (and  probably  by  the  latter  also)  when  a]i])lied  to 
the  mass-quantities  reduced  so  as  to  be  purchased  simply  by  the 
same  sums  of  money,  if  thus  continued  through  the  method  ol 
a})proach.  Here  is  additional  (M)nfirmation  that  in  such  cases 
the  common  result  so  reached  is  the  true  result.  ]5ut  in  other 
cases,  or  employed  without  such  reductions  or  corrections,  these 
two  purely  arithmetic  averages  diverge  and  scatter  their  results 
more  than  do  the  three  at  least  partly  geometric  methods.® 
Hence  the  latter  (or  at  least  two  of  them)  are  probably  nearer 
the  truth  than  either  of  the  ])urely  arithmetics  methods. 

Now  it  may  be  conceded  that  the  classes  with  rising  ])rices  and 
the  classes  with  falling  i)rices  may  freciueiitiy,  or  in  the  long 
run  generally,  be  nhout  even.      Therefore  the   purely  arithmetic 

'■'  And  its  iil)()V('  iioticc'fl  ill  Noti-  4  f^ciicrally  (ui  opposite  sidos  of  tlie  tliive. 


A    (JENERAL    TE8T    (^ASE  423 

methods,  if  |)n)}K'rly  corrected  through  the  inetliod  of  approach, 
would  generally  yield  results  almost  as  good  as  those  given  by 
the  partly  geometric  methods.  Yet,  to  repeat,  the  use  of  those 
methods  with  such  correction  is  impracticable. 

But  even  without  attempting  such  laborious  correction,  when, 
in  practice,  many  classes  are  dealt  with,  and  the  variati(ms  both 
of  prices  and  of  mass-quantities  are  such  moderate  ones  as  gen- 
erally take  j^lace  from  year  to  year,  containing  many  sources  of 
compensation,  and  the  total  money-values  of  the  classes  not  vary- 
ing much,  nor  the  general  exchange- value  of  money,  then  it  is 
probable  that  8cropc's  method  applied  to  the  arithmetic  means 
will  deviate  but  slightly  from  Scrope's  method  aj)plied  to  the 
geometric  means,  and  that  Professor  Lehr's  method  will  deviate 
but  slightly  from  the  method  with  double  weighting  aboye  ad- 
vocated. 

§  6.  For  the  purpijse  of  illustrating  these  inductions,  and  also 
of  casting  a  parting  glance  at  other  methods,  we  may  use  a  more 
complex  suppositional  example  than  any  used  as  yet.  In  order 
to  magnify  the  errors,  we  may  suppose  very  great  variations 
both  of  prices  and  of  mass-quantities  ;  and  also,  to  give  oppor- 
tunity for  the  working  of  compensatory  errors,  we  may  extend 
the  comparisons  over  a  short  series  of  periods.  Let  us,  then, 
posit  the  following  schema  : 


I  6  A  @  2 

12B@3 

10C@1 

3D@8 

6E@4 

1  F  @  6, 

II  4  A  @  3 

10B@7 

6C@  2 

4D@5 

6E@3 

2F@2, 

III  5  A  @  3 

13B@4 

8C@2i 

3D@4 

7E@3 

3F@2, 

IV  6  A  @  4 

12B@6 

9C@2 

2D@6 

5E@5 

2F@4, 

V  7  A@2 

11  B  @  7 

7C@3 

5D@3 

8E@4 

3F@3, 

VI  6  A  @  2 

12B@3 

10C@1 

3D@8 

6E@4 

1  F@6. 

Here  we  have  five  periods  with  different  states  of  things,  be- 
tween which  are  four  different  sets  of  variations.  In  addition 
it  is  supposed  that  at  a  sixth  period  everything  returns  to  exactly 
the  same  state  as  at  the  first.  The  object  in  making  the  last 
supposition  will  appear  presently. 

Measuring  every  set  of  ])rice  variations  separately  by  the  three 
methods  above  advocated,  we  obtain  the  following  percentages 
(the  positive  indicating  rises  and  the  negative  falls) — by  tlie 
geometric  method  : 


424  THE    r XI  VERSA  L    METHOD 

+  31.79,  -23.02,  +40.05,  -0.18,  -24.01;  (1) 
bv  the  method  with  double  weighting  : 

+  31.72,  -  22.92,  +  39.98,  -  0.30,  -  23.04;  (2) 
by  Scrope's  emended  method  : 

+  31.11,      -23.00,      +39.93,     -0.35,     -23.39.      (3) 

These  yield  the  following  index-numbers,  the  cohmms  repre- 
senting the  lines  similarly  numbered  : 


(1) 

(2) 

(3) 

T 

100 

100 

100 

II    ' 

131.79 

131.72 

131.11 

III 

101.45 

101.53 

100.95 

IV 

142.10 

142.02 

141.27 

^' 

133.29 

132.98 

132.30 

VI 

101.29 

101.54 

101.35 

Now  by  the  arrangement  of  the  prices  and  of  the  mass-quan- 
tities at  the  sixth  period  we  know  that  the  index-number  for 
the  sixth  period  ought  to  be  100.  Our  methods,*  then,  err  at 
the  sixth  period  by  between  1^  and  IJ  per  cent,  above  the 
truth. ^^  If  they  contained  a  single  sort  of  error  which  gradu- 
ally accumulated  at  every  advance,  we  might  correct  the  pre- 
ceding figure  by  reducing  it  by  four  fifths  of  1^,  IJ  or  1|  per 
cent.,  according  to  the  method  that  is  being  corrected,  and  the 
next  preceding  by  reducing  it  by  three  fifths,  and  so  on.  But 
this  we   cannot  do,  because  we   know  that  the  errors  in  these 

'"  Rather  furiously  it  is  with  weighting  according  to  the  arithmetic  means  of 
tlic  full  money-values  that  the  geometric  average  of  the  price  variations  gives,  at 
the  sixth  period,  in  this  example,  the  best  result,  its  indications  then  are  of  the 
following  percentage.'; : 

+  33.17,     -22.94,     +40.12,     -6.3.'?,     -25.10; 
forming  these  index-numbers : 

100,       1.33.17,        102.60,        143.S0,       134.69,      100.80. 
With  weighting  according  to  Gauss's  means  (see  Note  3  at  the  end  of  Sect.  III. 
above)  the  geometric  average  gives  these  indications: 

+  32.48,     -22.98,     +40.1.3,     -6.26,     -24.60; 
100,        132.48,       102.04,        143.00,      134.04,      101.07. 
With  weighting  according  to  the  arithmetico-geometrico-harraonic  means  (there 
also  suggested),  the  results  are  nearly  the  same,  being 

+  32.47,         22.9!),     +40.0!i,         6.24,      -  24..")7 ; 
100,        1.32.47,       102.0:!,        142. !i;'.,        1.34.01,        1(11. OS. 


A  (;kxki{al  ti:st  cask  425 

methods  are  not  ciiiuulative,  but  alternating.  The  fact  that 
these  methods  show  errors  above  the  truth  by  certain  amounts  at 
the  sixth  period,  gives  us  no  hint  as  to  tlie  errors  at  any  of  the 
preceding  periods.  For  instance,  we  (cannot  know  whether  the 
indications  given  for  the  fifth  period  are  above  or  behjw  the 
truth.  In  obtaining  the  sixth  index-number  from  the  fifth,  w^e 
have  measured  a  variation  of  the  general  price-level  from  the 
fifth  to  the  sixth  period,  which  is  just  the  inverse  of  the  Avhole 
variation  from  the  first  to  the  fifth  period.  Hence  to  test  the 
fifth  period  by  the  sixth  period  would  be  the  same  as  to  test  the 
serial  method  of  reaching  the  result  for  the  fifth  ])eriod  by  the 
direct  method  of  comparing  the  fifth  period  immediately  with 
the  first — which  latter  method,  as  already  said,  has  no  better 
claim  upon  our  approbation  than  the  former. 

Nor  w^ould  such  a  test  be  better  than  a  test  by  comparing  the 
fifth  period  directly  with  the  second.  In  fact,  if  we  use  such 
direct  comparisons  at  all,  we  ought  to  use  all  possible  ones. 
^A^e  ought  to  compare  not  only  every  period  with  the  first,  but 
every  period  with  the  second,  every  period  with  the  third,  and 
so  on.  In  the  following  table  the  index-numbers  resulting  from 
such  comparisons  are  stated,  the  method  with  double  weighting 
alone  being  employed  : 


Every  period 

compared  with 

the  tirst. 

Every  period 

compared  with 

the  second. 

Every  period 

compared  with 

the  third. 

Every  period 

compared  with 

the  fourth. 

Every  period 

compared  with 

the  tifth. 

1 

100. 

75.91 

94.42 

66.48 

76.36 

II 

131.72 

100. 

129.73 

91.83 

98.88 

III 

105.90 

77.08 

100. 

71.43 

79.97 

IV 

150.40 

108.89 

1.39.98 

100. 

106.79 

\- 

130.95 

101.13 

125.04 

93.64 

100. 

YI 

100. 

75.91 

94.42 

66.48 

76.36 

For  better  comparison  these  figures  may  be  re-arranged  as  follows, 
in  the  same  order. 


I 

100. 

100. 

100. 

n 

131.72 

131.72 

137.39 

III 

105.90 

101.53 

105.90 

IV 

150.40 

143.44 

148.25 

\' 

130.95 

13,3.22 

132.42 

VI 

100. 

100. 

100. 

As 

no  one   of 

these  series 

is   pref( 

100. 

100. 

138.13 

129.49 

107.44 

104.72 

150.40 

139.85 

140.84 

130.95 

100. 

100. 

to   another,  we   may 


426  THE    UNI  VERS  AI.    METHOD 

average  them, — doing  so  on  the  iirst  table,  doubling  the  tirst 
column  in  order  to  represent  also  the  comparisons  with  the  sixth 
period,  and  using  the  arithmetic  average  for  want  of  a  better. 
The  series  thus  obtained  is  as  follows : 

100,     133.26,     105.28,     145.46,     132.84,     100. 

Yet  such  a  series  cannot  be  accepted  as  authoritative. 

Nor  are  better  results  obtained  by  using  these  methods  ad- 
apted to  a  whole  epoch — say  of  the  five  different  periods.  The 
geometric  method,  with  weighting  according  to  the  geometric 
averages  of  the  full  money-values  of  the  classes  over  all  the  five 
periods,  yield  these  index-numbers  : 

100,     140.0,     109.5,     155.7,     141.9,     100.^' 

The  method  with  double  weighting  applied  to  the  geometric 
averages  of  the  prices  of  the  mass-units  over  all  the  five  periods, 
these  : 

100,     131.01,     106.02,     149.49,     135.12,     100.'' 

Scrope's  method  applied  to  the  geometric  averages  of  the  mass- 
quantities  over  all  the  five  ])eriods,  these  : 

100,     130.12,     104.38,     141.04,     133.93,     100.'^ 

None  of  these  is  satisfactory.  It  is  ])lain  that  if  another  period 
were  included  in  the  epoch  at  the  beginning  or  at  the  end,  all 
the  indications  would  be  shifted.  Thus,  for  example,  with  the 
sixth  period  added  to  the  epoch,  the  method  with  double  weight- 
ing aj^plied  to  the  geometric  averages  of  the  prices  over  all  the 
six  periods  yields  these  index-numbers  : 

100,     134.49,     105.07,      149.76,      132.15,      100.'* 

No  correction  of  the  figures  first  given,  therefore,  seeuis 
to  be  ])()ssible.     All  that  we  can  do  is  to  take  them  as  approxi- 

''  AVith  weigliting  according  to  the  arithmetic  averages  : 

100,  140.3,  109.0,  l.M.S,  142.3,  100. 
^2  Applied  to  tiie  arithmetic  averages  : 

100,  136.77,  105.89,  14G.r.7,  135.12,  100. 
1^  Ajjplifd  to  tlie  arithmetic  averages  : 

100,  128.76,  100.17,  140.80,  132.43,  100. 
' '  A])])lifd  to  tlie  arithmetic  averages  : 

Kill,      I4II.1S,     l()4.!t5,      149.13,  141. 2S,      KlO. 


A    GENKKAL    TKST    CASE  427 

matoly  representing  the  trne  variation   in  the  general   level  of 
prices. 

§  7.  Let  ns  now  turn  to  the  indications  yielded  by  other 
methods  applied  to  this  same  schema,  beginning  with  those 
which  make  most  pretension  to  accuracy. 

Professor  Lehr's  method  gives  the  following  percentages  of 
the  separate  price  variations  : 

+  31. ()9,       -  22.86,       +  39.84,       -  5.70,       -  28.09; 

which  yield  these  index-numbers  : 

100,        131. 09,        101.58,        142.05,        133.9(3,        96.33. 

These  are  very  close  to  those  given  by  the  three  methods 
similarly  used  in  serial  form,  except  only  in  the  last  indication. ^^ 
Scrope's  method  presents  many  varieties.  We  have  seen  that 
two  ways  of  drawing  a  mean  between  the  mass-quantities  at  the 
two  periods  have  jireviously  been  recognized.  One  of  these  is 
U)  apply  it  to  the  arithmetic  means  of  the  mass-quantities. 
This  gives  the  following  percentages  : 

+  80.80,     -23.00,      +40.04,     -6.96,     -22.43.     (1) 

Another  is  to  make  the  calculations  once  on  the  mass-quantities 

'^  The  method  above  suggested  in  Note  9  in  Sect.  II.  gives  these  percentages  : 
+  31.71,     -23.06,     +40.26,     -6.58,     -24.09, 
which  yield  index-numbers  as  follows  : 

too,      131.71,      101.34,      142.14,      132.78,      100.79, 
which  also  are  very  close.     On  the  other  hand,  the  method  suggested  in  Appendix 
C,  VI.  ?  3  as  a  possible  form  of  Nicholson's  method  gives  the  following  percent- 
ages : 

+  33.91,     -12.95,     +55.70,     -9.10,     -2.41, 

which  yield  these  index-numbers : 

100,      133.91,      116.57,       181.50,      164.98,      160.99, 
—the  worst  of  all.     The  metiiod  (with  geometric  average  of  the  inverted  mass- 
(juautity  variations)  there  also  (in  Note  4)  suggested  as  a  variant  upon  the  last, 
would  give  these  percentages  : 

+  23.83,     -22.17,     +48.00,     -19.57,      -12.86, 
forming  these  index-numbers : 

lf«1,  123.83,  96.38,  142.64,  114.72,  KW, 
which  are  better,  but  still  not  good.  It  may  be  noticed  that  it  is  impossible  to 
apply  Drobisch's  method  to  this  example  without  further  specifying  the  weights 
(or  capacities)  of  the  mass-quantities.  Various  suppositions  wlth-Yegard  to  these 
would  make  Drobisch's  method  yield  various  results,  but  would  have  no  influence 
upon  the  results  given  by  any  of  the  other  methods. 


428  THE    UNIVERSAL    METHOD 

of  the  earlier  periods  and  again  ou  the  mass-quantities  of  the 
later  periods,  and  to  average  the  results.  Thus  Scrope's  method 
on  the  mass-quantities  of  the  earlier  period  in  every  comparison 
(the  same  as  the  arithmetic  average  of  the  price  variations — 
Young's  method — with  the  full  weighting  of  the  earlier  period) 
gives  these  percentages : 

+  40.18,     -22.79,     4- 42.06,     -1.29,     -14.29.     (2) 

And  Scrope's  method  on  the  mass-quantities  of  the  later  period 
in  every  comparison  (the  same  as  the  harmonic  average  of  the 
price  variations  with  the  full  weighting  of  the  later  period), 
these : 

+  21.43,     -23.17,     +37.78,     -11.58,     -30.87.    (3) 

The  arithmetic  means  between  these  percentages  are  the  follow- 
ing percentages  : 

+  30.80,     -  22.98,     +  39.92,     -  6.44,     -  22..58.       (4) 

In  stringing  these  results  out  in  a  series  of  index-numbers 
there  is  a  split  into  two  ways  of  forming  the  mean  series.  The 
one  is  to  form  the  index-numbers  on  the  (arithmetic)  mean  per- 
centages, as  given  in  line  (4).  The  other  is  to  draw  the  (arith- 
metic) means  of  the  index-numbers  in  the  two  series  formed  on 
lines  (2)  and  (3).     The  series  so  formed  is  added  in  column 


I 

(1) 

100. 

100. 

(3) 
100. 

(4) 
100. 

(5) 
100. 

II 

130.80 

140.18 

121.43 

130.80 

130.80 

III 

101.72 

108.22 

93.29 

100.74 

100.75 

IV 

142.45 

153.74 

128.53 

140.96 

141.13 

V 

132.53 

151.76 

113.65 

131.88 

132.70 

VI 

102.80 

130.07 

78.57 

102.10 

104.82. 

The  results  in  the  average  methods  (in  columns  1,  4  and  5)  are, 
like  those  of  Professor  Lehr's  method,  very  close  to  those  given 
by  the  three  more  theoretically  correct  methods,  showing  that 
in  practice  all  these  methods  are  likely  to  give  results  very 
nearly  alike.  The  figures  in  columns  (2)  and  (3)  are,  singly, 
extravagant  and  absurd.  But  there  is  order  in  their  extrava- 
gance ;  for  the  nearness  of  their  means  to  the  more  truthful  re- 


A    (iKNEUAL    TEST    CASE  429 

suits  show.s  that  tliey  straddle  the  true  course,  the  one  varying 
on  the  one  side  about  as  tlie  other  does  on  the  other.""'  This  is 
as  we  might  expect  from  reasoning  d  priori ;  for  we  liave  seen 
that  the  one  has  no  more  reason  for  it  than  tlie  other,  and  there- 
fore the  one  has  no  more  reason  against  it  than  the  other.  Be- 
tween the  various  ways  of  drawing  the  mean  there  is  little  choice, 
except  that  the  first  is  the  most  convenient.'^ 

As  for  Young's  method,  when  it  is  employed  with  Avhat  we 
have  found  to  be  proper  weighting,  though  not  })roper  for  it, 
this  gives  most  absurd  results.  Thus,  employed  with  weighting 
according  to  the  geometric  means  of  the  full  money-values  at 
both  periods,  its  indications  for  the  price  variations  are 

-f- 55.66,      -19.76,        -f- 4:5.62,        +1.44,        —0.58, 

forming  these  index-numbers  : 

100,         155.66,        124.90,         179.38,        181.97,        180.91. 

Or  with  weighting  according  tft  tlie  arithmetic  means,  the  per- 
centages are 

+  57.12,     -19.35,       +43.67,       -0.57,       -1.8.S, 

^''Of  course  the  i^ roper  mean  to  draw  here  is  really  the  geometric.  The  per- 
centages of  the  geometric  means  between  the  variations  whose  percentages  are 
given  in  lines  (2)  and  (3)  are 

+  30.47,     -22.92,     +39.90,     -6.58,     -23.02 
which  yield  these  index-numbers  : 

100,  130.47,  100.48,  140.57,  131.23,  101.10. 
1'  We  might  also  use  the  geometric  average  of  the  price  variations  in  the  way 
the  arithmetic  has  just  been  used — twice  applying  it  to  the  same  price  variations, 
once  with  the  weighting  of  the  earlier,  and  once  with  the  weighting  of  the  later 
period,  and  then  drawing  the  geometric  mean  between  the  two  results.  This 
yields  the  following  percentages — of  the  price  variations  on  the  weighting  of  the 
earlier  periods : 

+  16.94,     -26.01,     +38.11,     -7.25,     -33.03; 

of  the  price  variations  on  the  weighting  of  the  later  periods : 

+  48.12,     -19.49,     +41.80,     -5.47,    -11.58; 
of  the  geometric  mean  of  these  variations  : 

+  31.61,     -22.82,     +39.94,     -6.36,     -23.05; 
and  the  following  index -numbers  on  tlie  last  percentages  : 

100,       131.61,       101.58,       142.-15,       133.11,       102.42. 
But  these  are  no  better  than  those  directly  obtained  by  the  geometric  average 
with  the  geometric  means  of  the  weights  of  l)oth  periods. 


430  THE    UNIVERSAL    METHOD 

and  the  index-numbers 

100,         l.")7.12,        12().7j!,         IS-J.o."),         lsl.()2,        177. Gl. 

Even  with  the  weighting  of  the  first  ])eri()d  alone  in  every  com- 
parison,— weighting  which  is  proper  for  the  arithmetic  average 
of  the  price  variations  when  the  mass-quantities  are  constant, — 
this  method  still  errs  largely,  although  it  is  now  much  better 
than  Mith  mean  weighting.  The  figures  have  already  been 
given — under  Scrope's  method,  with  which,  in  one  form,  this 
method,  so  weighted,  is  identical. 

Furthermore  there  is  the  method  of  using  the  geometric 
average  of  the  price  variations  with  the  weights  that  are  the 
smaller  at  either  of  the  two  periods  compared,  and  the  variety  of 
Scrope's  method  which  applies  it  to  the  smaller  of  the  mass- 
quantities  at  either  of  the  periods  compared.  The  former  gives 
these  percentages  and  index-numbers  : 

+  26.30,     -23.24,      +39.3:5,     -  l.HO,       -19.72, 
100,         12().30,         9().9r),         13r).()8,        132.92,  106.71; 

and  the  latter  these  : 

+  ;>,!. 0;>,,      —22.90,      +39.09,      -3.40,         -28.09, 

100,       131.63,         101.16,         141.16,  136.29,  97.92. 

These  figures,  although  sometimes  nearly  right,  are  erratic, 
and  untrustworthy.  Nor  is  anything  trustM'orthy  reached  by 
taking  an  average  between  them. 

§  8.  Lastly  it  remains  to  see  what  plan  should  be  adopted 
when  we  are  reviewing  the  course  of  the  exchange-value  of 
money  in  times  past,  for  which  complete  and  accurate  data  are 
not  obtainable.  To  attempt  to  use  the  complex  methods  above 
recommended  as  the  best  would  be  pedantic ;  for  the  figures 
posited  in  our  example,  (^specially  of  the  mass-quantities,  are  now 
to  be  taken  as  only  approximately  correct.  We  must  still  use 
uneven  weighting  :  but  it  can  only  be  rough  uneven  weight- 
ing. There  are  two  separate  and  distinct  systems  ready  for  our 
adoption. 

We  can  either  (1)  draw  some  average  of  the  total  money- 
values  of  the  classes  during  an  ejwch  of  years,  and  with  weighting 


A    (illXKUAL    TKST    CASK  431 

SO  determined  employ  the  geometric;  a\'eraj»,e  ot"  the  priee  vari- 
ations ;  or  (2)  draw  some  averajj^e  of  tlie  mass-quantities  of  the 
classes  during  the  ejxx'h,  and  ap|)ly  to  them  Scrope's  method. 
In  eaeh  case,  in  getting  the  avei'age  weights  or  mass-<|uantities, 
we  might  as  well,  f(>r greater  convenience,  employ  the  arithmetic 
average,  because  in  thes(>  complex  instances  there  is  little  differ- 
ence in  the  results  whether  we  use  the  aritiimetic  or  the  geometric 
average,  and  neither  is  exactly  true.  But  If  ive  use  an  average 
of  the  total  moneij-vahies  for  our  iveiglitlmj ,  we  mud  use  the  geo- 
metric  average  of  the  priee  variations  ;  and  if  we  use  an  average 
of  the  mass-quant  it  ie.'<,  we  mud  uxe  Scrope's  method.  There  must 
be  no  intermingling  of  these  operations. 

Both  these  methods  so  used,  we  have  seen,  universally  satisfy 
Professor  Westergaard's  test.  Hence  it  is  indifferent  whether 
we  compare  the  periods  successively,  or  compare  each  succeeding 
period  with  the  same  original  base  (whether  at  the  beginning, 
end,  or  in  the  middle,  of  the  series).  The  calculations  whereby 
the  rough  weighting  is  obtained  should  be  renewed  from  time  to 
time — say  in  every  decade.  This  will  require  the  starting  of 
new  chains,  from  new  bases.  But  the  results  may,  if  desired, 
all  be  strung  out  in   one  series. 

On  our  schema  these  two  methods  have  already  been  worked 
out  with  the  exact  weighting,  and  applied  to  the  exact  averages 
of  the  mass-quantities,  over  all  the  five  periods.''^  The  results 
so  obtained  by  Scrope's  method,  in  this  particular  instance,  are 
better  than  those  obtained  by  the  geometric  method  ;  and  in 
other  cases  we  may  be  pretty  sure  they  will  be  as  good,  if  not 
better.  Scrope's  method  is,  then,  recommended  by  its  greater 
convenience.  Over  the  five  periods  in  (piestiou  the  exact  arith- 
metic averages  of  the  mass-(piantities  are:  5|  A,  11|  B,  8  C, 
32  D^  (^2  E^  2^  F.  If  in  their  places  we  used  only  rough 
averages,  as  e.  g.,  in  the  same  order,  5,  11,  8,  3,  6,  2,  applied 
to  these  Scrope's  method  would  give  the  following  series  of 
index-numbers  : 

100,  127.77,        100.1)2,         140.74,         132.40,  100, 

1"  Ahove  ill  Notfs  12  and  13. 


4'.l'2  THE    UNIVERSAL    METllol) 

wliicli,  it  will  l)e  noticed,  are  still  fairly  aeeiirate.  But  there 
are  occasions  when,  havinti'  a  general  idea  of  the  relative  im- 
portance of  the  classes,  we  may  more  conveniently  nsc  the  geo- 
metric average  of  the  price  variations  weighted  accordingly. 

That  the  distinctive  elements  in  these  methods  should  not  be 
mixed  together,  in  violation  of  our  above  canon,  may  be  shown 
bv  the  results  obtained  from  the  a})])lieation  of  Young's  method 
to  this  schema.  The  following  index-numl)ers  are  obtained  by 
arithmetically  averaging  the  price  variations  at  the  successive 
periods,  all  compared  with  the  first,  in  every  comparison  with 
the  same  weighting,  which  is  the  arithmetic  average  of  the  total 
money-values  over  the  five  periods  : '" 

100,  ]()3.()<),         12;i.93,  IHo.ST,  172.16,        100, 

— the  extravagance  of  which  runs  to  absurdity. 

By  the  same  schema  may  be  tested  a  fe^v  m(»re  methods  that 
have  been  put  into  practice.  Jevons's  method,  also  employed 
by  Walras  and  Simon,  using  the  geometric  average  of  the  price 
variations  with  even  weighting,  gives  these  index-numbers  : 

100,  101.50,  1)2.47,  i;50.7(),  104.(>;),        100, 

indicating  these  percentages  of  variation  between  the  periods  : 

_^1. ;-)(),         -8.90,         +41.40,       —  1U.9S,         -4.41. 

And  Carli's  method,  as  extended  by  Evelyn,  and  as  employed, 
for  instance,  in  the  Economist,  drawing  the  arithmetic  average 
of  the  price  variations  with  even  weighting,  gives  the  index- 
numbers  : 

100,  125.70,  115.2S,  144.44,  i;i0.80,        100, 

indicating  these  ])ercentages  of  variation  between  the  periods  : 

+  25.70,        —  8.2!»,       +  25.;]0,       —  5.21),         —  2G.1»0. 

Thus  in  this  exam])le,  as  Jevons  himself  claimed,  his  method 
gives  lower  results  than  the  ordinary  arithmetic  method.  But 
its  results  are  more  alx^rrant  even  than  those  given  by  the  latter 
method. 

'■^  TIk'  saiin'  wc'iglitiiiK  as  aliovc  used  uitli  llu'  geoiiK'tric  average  of  the  price 
variations  in  Note  11. 


riiK   MissiN(;    i'i;()i'(»siii(»Ns  sri'ri.ii:it  4:);> 

The  correction  of  the  hitter  method  ofl'ered  hy  Mr.  Palgrave, 
arithmetically  averaging  the  ])rice  variations  of  the  later  periods 
compared  with  the  first,  in  every  comparison  w  ith  the  weighting 
of  the  later  period,  appears,  in  this  exanij)le,  worse  than  what  it 
Avonld  im])rove  upon  ;  foi'  its  index-numbers  are 

100,  171.07,  1:51.-11,  170.05,  174.08,         jOd, 

which  are  com])aral)le  only  with  those  above  given  by  Young's 
method,  of  which  this  method  itself  is  a  variety.  In  fact, 
Young's  method,  in  every  form,  has  been  found  to  be  bad.-" 

It  must  be  remembered,  however,  that  in  actual  })ractice  the 
deviations  of  these  methods  would  be  much  smaller  than  they 
are  in  this  example. 

Y. 

§  1 ,  As  a  risume  of  the  little  which,  after  all  these  investiga- 
tions, we  have  learned  with  absolute  exactness  and  positiveness, 
we  can  now  supply,  in  part,  two  propositions  that  have  been 
lacking  hitherto. 

The  one  of  these,  which  was  missed  in  Chapter  VII. ,  is  in  re- 
gard to  compensatory  price  variations.  Because  of  the  perfect 
accuracy  of  the  geometric  average  with  even  weighting,  or  the 
geometric  mean,  of  the  price  variations  in  all  cases  dealing  with 
two  classes  between  whose  full  weights  at  each  of  the  two 
periods  compared  the  geometric  means  are  equal,  we  have  this  : 
Of  two  clas.'ies  equallij  important  over  both  the  periods  compared 
compensator t/  price  variations  are  siin2)le  geometric  variations, 
that  is,  variations  from  nnity  to  the  opposite  geometric  extremes 
around  nnity,  so  that  the  geometric  mean  between  them  is  unity, 
indicative  of  constancy  (Proposition  XLVIL).  In  other  words, 
if  the  price  of  [A]  rises  by  p  ])er  cent,  to  1  -|-  p  times  its  former 
price,  in  order  that  the  exchange-value  of  money  remain  con- 

stant,  the  price  of  fBI  must  fall  bv per  cent,  to  ^     -      of 

'  1  LJ  •    I  ^  p  J^  1  +  p 

its  former  price ;  and  reversely  a  fall  of  [A]  by  p'  per  cent,  to 

-"With  such  weighting  so  used  tlie  harmonic  average  of  the  price  variations 
yields  better,  but  still  unsatisfactory,  results,  its  index-numbers  being 

100,      i47.ti2,      00.21,      ]r)0.4s,      ii(;.(ir),      lOO. 
28 


434  Tin;  imvkksal  mkthod 

1  —  y/  of  its  foriiKM'   i)ri('e  would  he  conipciisatcd   hy  a    rise  of 

V'  1         .  .      ,. 

[B]   1)V  Y ,  ])('!•  cent,  to  ,  tunes  its  tornicr  ])riee, — jiro- 

vided  the  aeeunipauyinji,-  states,  or  variations,  of  the  niass-c^nan- 
tities  are  siicli  as  to  make  these  classes  equally  important  over 
both  the  periods  (which  variations,  if  they  he  of  the  mass-quan- 
tities together  purchased  at  each  [)eriod  with  the  same  t<5tal  sum 
of  money,  will  likewise  be  found  to  l)e  sim])le  geometric  vari- 
ations). This  proposition  is  true  in  a  world  with  only  two 
classes  of  commodities  beside  money,  if  the  classes  so  vary  in 
price  (and  in  mass-quantities).  It  is  true  also  in  a  world  wdth 
any  number  of  classes  of  commodities,  referring  to  two  of  them, 
provided  the  others  do  not  vary  in  price,  or  vary  in  such  wdse 
that  they  compensate  for  one  another  (in  pairs,  or  otherwise) 
and  do  not  cause  an  alteration  in  the  exchange-value  of  money 
(according  to  Proposition  XXXI.).  These  others  being  such, 
their  numbers  will,  as  we  have  seen  (in  Proposition  XIX.), 
cause  the  general  exchange- value  of  money  to  fall  less  through 
the  single  rise  in  price  of  the  one  class  than  it  would  fall  with- 
out them  ;  but  they  will  cause  the  general  exchange-value  of 
money  to  rise  equally  less  through  the  single  fall  in  price  of  the 
other  class.  And  the  influence  of  these  price  variations  singly 
to  deflect  the  general  exchange-value  of  money  being  reduced  in 
the  same  proportion  (according  to  Proposition  XXV.),  the  com- 
pensation is  the  same,  no  matter  how  numerous  the  other  classes 
may  be,  or  how  large.  Or,  if  the  other  classes  have  varied  so 
as  to  cause  a  variation  in  the  general  exchange-value  of  money, 
the  influence  of  these  two  classes  is  such  as  it  woidd  be  if  their 
prices  had  remained  constant.  This  last,  however,  belongs  to 
the  next  J*r()|)osition. 

But  we  cannot  extend  this  Proposition  so  as  to  say  that 
among  three  equally  important  classes  (over  both  the  periods 
compared)  the  rise  of  [A]  by  p  per  cent,  to  1  -f-  ^  times  its 
former  price  is  compensated  by  a  fall  of   [B]  and  [C]  each  by 

a  percentage  carrying  its  ]>rice  down  to  of   its    former 

l)riee,  that  is,  to  the  figure  between  which  twice  ivpeated  and  the 


THE  missinm;    i-uoi-osiiions  si:i'1'1Ji:i)  435 

other  the  geometric  (trcnu/c,  witli  even  \\('i*rhtiii<i-  foi-  tlie  three 
classes,  is  iinitv,  indicative  of  constancy  ;  for  this  is  not  true, — 
and  yet  it  is  very  near  the  truth.  Xor  can  we  extend  it  to  any 
more  complex  cases,  or  to  cases  with  uneven  weightino;.  Yet 
in  all  these,  with  the  amount  of  unevenness  in  wei<»-htinj>-  and  of 
variation  in  prices  likely  to  l)c  met  with  in  practice,  the  state- 
ment that  compensatory  price  variations  must  be  such  that  the 
geometric  average  of  them,  with  projxM"  weighting,  should  be 
unity,  is  likely  to  be  nearly  correct.  The  ])eculiaritv  of  the 
geometric  average  must  always  be  borne  in  mind,  that,  while 
the  geometric  mean  proper,  between  only  two  cipially  important 
classes,  and  therefore  with  even  weighting,  in  all  cases  where  it 
can  rightly  be  used,  yields  the  true  result,  the  geometric  average, 
between  more  than  two  classes,  or  between  unequally  important 
classes,  wherever  uneven  weighting  has  to  be  emjdoyed,  yields 
a  result  deviating  above  or  below  the  truth  according  to  what 
seem  to  be  fixed  laws,  but  only  slightly  erring  in  ordinary  cases. 
Wherever  in  j)ractice,  among  a  large  number  of  classes,  some 
large  and  some  small,  some  rising  and  some  falling  in  j)rice,  the 
geometric  average  with  its  pro})er  weighting  indicates  constancy, 
we  have  good  reason  to  believe  that  the  true  answer  Avould  show 
at  most  but  a  trifling  variation. 

The  other  Proposition,  which  was  missed  in  Chapter  VI.,  is 
more  comprehensive, — in  fact,  it  embraces  the  preceding.  It  is 
this  :  Wlun  of  tiro  r/r/.swN  e(/U((/lt/  iinporfaiit  over  both  tlie  pcrioih 
compared^  the  p7'ice  of  the  one  reinaiiix  .sttttioiiar;/  whi/e  that  of  the 
other  varies,  or  the  prices  of  both  vari/,  the  infJaenee  of  these  p)vice 
variations  upon  the  general  exchange-value  of  inonei/  is  the  same  as 
it  would  be  if  both  the  clasxes  varied  (dike  in  the  geometric  mean 
between  their  actual  price  variationx  (Proposition  XLVIII.). 
This  is  the  only  definite  answer  that  can  universally  be  given  to 
Jevons's  problem,  to  which  a  threefold  answer  was  given  in  the 
last  Chapter,  on  the  condition  there  posited  that  the  mass-quanti- 
ties are  constant  over  both  the  periods.  The  present  Projiosition 
is  applicable  whenever  the  geonietric  mean  between  the  full 
weights  of  one  class  at  the  two  })eriods  is  the  same  as  the  geo- 
metric mean  between  the  full  weights  of  one  othei-  class,  what- 


436  THE    rXIVEltSAL    METHOD 

ever  be  the  variations  of  the  mass-quantities,  provided,  of 
course,  they  be  such  as  with  the  price  variations  to  permit  of 
equality  between  the  means.  It  directly  covers,  therefore,  all 
the  cases  in  (Chapter  X.  in  which  equal  sums  are  supposed  to  be 
constantly  sjjcnt  on  two  classes  at  both  periods.  It  covers  also 
those  cases  in  Chapter  XI.  in  which  the  mass-quantities  are 
supposed  to  be  constant,  provided  the  sums  spent  at  each  period 
have  this  property  of  equality  between   tlieir  geometric  means. 

But  again  this  Proposition  cannot  be  extended  to  permit  us 
to  say  that  the  influence  of  the  ]>rice  variations  of  any  tAvo 
classes,  or  of  any  number  of  classes,  is  the  same  as  it  would  be 
if  they  all  changed  alike  in  the  geometric  average  between  their 
actual  price  variations  weighted  in  the  manner  always  pre- 
scribed. And  yet  in  most  cases  their  influence  would  be  very 
nearly  such.  Again  what  is  true  of  the  geometric  mean  is  not 
exactly  true  of  the  geometric  average. 

In  the  absence  of  an  absolutely  true  method  of  measuring 
variations  of  general  exchange-value,  it  is  not  convenient  to  im- 
prove upon  the  ])recediug  general  statements  by  attempting  to 
formulate  propositions  in  accordance  Avith  either  of  the  other  two 
methods  of  measurement  which  we  found  reason  to  believe  to  be 
slightly  better  than  the  geometric  method.  Yet  by  means  of 
the  formulse  for  these  methods  the  influence  of  the  price  vari- 
ations of  any  numbers  of  classes,  or,  given  certain  price  varia- 
tions, the  compensatory  price  variations  of  any  numbers  of  other 
classes,  all  the  other  conditions  being  stated,  is  easily  found 
with  still  closer  approximation  to  the  truth. 

§  2.  If,  however,  it  ever  happens,  or  if  we  suppose,  tliat  tlie 
mass-quantities  are  constant  through  both  the  })eriods,  thus  cov- 
ering all  file  cases  in  Chapter  XI.,  it  is  easy  to  sunnnarize  the 
])rincij)les  there  discovered  and  to  state  the  lacking  Pro])ositions, 
confined  to  these  special  cases,  in  twofold  forms  Avith  j)erfect 
definiteness  and  universality.  The  first  is  :  With  eonstant  //i«,ss- 
quantities,  compensatory  price  variations  are  arithmetic  variations 
with  the  weighting  of  the  first  period,  or  harmonic  variations  with 
the  weighting  of  the  second  period  (Proposition  XLIX.).  The 
second :     With   constant   mass-quantities,  the   influence   upon  the 


TUK  ^rIS8rN(;   i'roi'ositions  suprLUOD  437 

general  exchange-value  of  inoiiey  of  ant)  two  or  more  price  varia- 
tions is  the  same  as  if  the  prices  all  varied  in  the  arithmetic 
average  loith  the  weighting  of  the  first  period,  or  in  the  harmonic 
average  with  the  imighting  of  the  xccoud,  period  (Proposition  L.), 
Tlu^  third  forms  in  which  these  Propositions  can  also  be  stated 
have  already  been  included  in  the  preceding  Propositions.  The 
last  two  Propositions  have  little  chance  of  application.  But 
what  we  know  about  Scrope's  emended  method  shows  that 
similar,  but  over-complex,  Propositions  could  be  framed,  applied 
to  the  geometric  means  (and  in  most  cases  even  to  the  arith- 
metic means)  of  the  mass-quantities  of  every  class  at  the  two 
periods,  that  would  very  closely  a})proximate  to  the  truth. 


CHAPTER    XIII. 

THE  DOCTRINE  OF  THE  CONSERVATION  OF  EXCHANGE- 
VALUE,    AND  THE  MEASUREMENT    OF   EXCHANGE- 
VALUE   IN   ALL  THINGS. 


§  1.  In  Chapter  II.  our  Proposition  XXXIX.  was  worded  : 
"  All  things  collectively,  provided  they  be  the  same  (or  similar 
and  m  equal  quantitiesj  at  all  the  periods  compared,  are  con- 
stant in  exchange- value."  Examination  of  this  princijile  was 
deferred.     We  are  now  prepared  to  examine  it. 

This  proposition  is  not  new.  In  one  form  or  another  it  has 
often  been  intimated.  Originally,  and  most  frequently,  it  has 
been  stated  in  the  form  that  it  is  impossible  for  all  things  to  rise 
or  fall  in  value.  It  was  so  stated  by  Senior  as  early  as  1836, 
and  by  J.  S.  Mill  in  1848,  after  the  latter  of  whom  it  has  con- 
stantly been  repeated.^  Sometimes  little  more  has  been  meant 
than  the  obvious  assertion  that  neither  all  commodities  together 
can  rise  in  exchange-value  nor  all  commodities  together  can  fall 
in  exchange-value.      But  usually  there  has  also  been  an  impli- 

^  Senior,  Politicul  economy,  p.  21  (originally  in  the  Encyclopaedia  Metropoli- 
tana,  1836 :  also  in  Arrivabene's  translation  of  Senior's  writings,  Paris  1836,  p. 
104);  Mill,  op.cit.,  Vol.  I.,  pp.  540,  588;  Levasseur,  R.  IS,  pp.  138,  158  ;  Courcelle 
Seneuil,  op.  cit.,  Vol.  I.,  pp.  259,  272;  J.  Bascom,  Political  economy,  Andover 
1860,  p.  224;  Fawcett,  op.  cit.,  p.  312;  Bolles  (quoting  Fawcett),  op.  cit.,  p.  55  ; 
Cairnes,  op.  cit.,  p.  12;  Macleod,  Theory  of  hanking.  Vol.  I.,  p.  70,  Theory  of 
credit.  Vol.  I.,  p.  176;  A.  L.  Perry,  Introduction  to  political  economy,  1881,  p. 
65;  Mannequin,  Question  nionetaire,  1881,  p.  7;  Jourdain,  Cours  d'iconomie 
politique,  1882,  p.  435  ;  Hansard,  B.  67,  p.  1 ;  R.  T.  Ely,  Introduction  to  political 
econow*?/,  1889,  p.  179  ;  S.N.  Patten,  Theory  of  dynamic  economics,  1892,  pp.  64-65, 
and  Cost  and  expense,  1893,  p. 57  (of  "objective  values");  Denis,  B.  125,  p.  171 ; 
Fonda,  B.  127,  p.  6;  Laughlin,  op.cit.,  pp.  147-148;  Bourguin,  B.  132,  p.  25; 
Parsons,  B.  136,  pp.  114,  115  (of  "internal  exchange-values,"  offering  a  dem- 
onstration vvliicli  is  niathcinaticiilly  iiicori'cet). 

438 


I'lii;  i)(H  ri;iNi':   i)i;s(  i;ii!i;i)  439 

cation  of  coiinterhalancing,^  with  the  meaning'  that  all  comnriod- 
ities  together  can  neither  rise  nor  fall  in  excliange-value — in 
other  words,  they  are  constant  in  exchange-value.^  In  this 
sense  it  is  true  only  if  the  commodities  are  the  same  (or  similar, 
in  equal  mass-quantities)  at  all  the  periods  compared.  This 
proviso  has  been  noticed  by  Cairnes,  who  added  that  "  the 
aggregate  amount  of  values  "  may  increase  or  decrease  with  an 
increase  or  decrease  of  commodities.^ 

The  need  of  this  proviso  may  be  made  a])])arent  by  analogy. 
If  we  have  a  universe  consisting  of  three  atoms  endowed  with 
equal  attractive  force,  the  attractive  ft)rce  of  each  atom  is  one 
third  of  the  Avhole  attractive  force  in  the  universe.  If  then  a 
fourth  atom  is  created  in  this  universe,  endowed  with  attractive 
force  equal  to  that  of  each  of  the  others,  although  the  force  in 
each  of  the  atoms  now  sinks  to  one  fourth  of  the  whole,  it  is  one 
fourth  of  a  whole  larger  by  one  third,  and  ^  x  (3  -f  1)  =  ^  X  3, 
so  that  each  of  the  three  atoms  continues  to  have  the  same  force 
as  before,  and  as  there  is  one  more  such  atom,  the  attractive 
force  in  the  universe  has  increased.  Similarly  if  we  have  an 
economic  world  consisting  of  three  articles,  exchanging  equally 
for  one  another,  so  that  they  are  endowed  with  equal  purchasing 
power  or  exchange-value,  the  purchasing  power  possessed  by 
any  one  of  these  is  one  third  of  the  whole  purchasing  power  in 
this  world.     Then  if  another  article  is  produced  in  this  world, 

-  Sometimes  erroneously.  Thus  Ely  :  "  Let  us  suppose  that  to-day  two  bushels 
of  wheat  exchange  for  three  of  oats,  and  to-morrow  for  four  bushels  of  oats.  We 
may  say  that  wheat  has  risen  in  value,  but  it  is  obvious  that  exactly  in  the  same 
proportion  oats  have  fallen  in  value,"  loc.  n't.  Wheat  has  risen  33§  per  cent,  in 
exchange-value  in  oats  ;  oats  have  fallen  2i>  per  cent,  in  exchange-value  in  wheat. 
These  proportions  are  not  the  same.  There  is  geometric  equality  of  proportion  ; 
but  this  should  be  specified. — A  mistake  like  this  is  responsible  for  Parson's 
attempted  demonstration. 

^  This  inference  has  been  directly  drawn  by  Fonda  (without  the  proviso),  H. 
127,  p.  18. — In  this  sense,  but  ambiguously  about  "value,"  the  proposition  has 
been  combated  by  A.  Clement,  De  /'influence  e.eercee  par  la  hausseou  la  baise  des 
valeurs  siir  la  richesse  generate,  Journal  des  Economistes,  July  1854,  pp.  8-10> 
because  he  perceived  that  use-values  can  rise  or  fall  together;  and  by  Pollard, 
op.  cit.,  Ch.  IX.,  because  it  is  not  true  of  cost-value,  in  which  sense  alone  he  pre- 
fers to  use  the  term  "value."  Cf.  also  Kicardo  above  (jiioted  in  Chapt.  I.  Sect. 
IV.  Note  8. 

*  Op.  cit.,  pp.  12-i;{.  The  proviso  has  also  been  incorporated  in  the  statement 
by  Manne(iuin,  loc.  cit. 


44<l  THE    rONSERYA'l  ION    OF    EXCII  A.\(;  K-VALUE 

whicli  will  excliuugc  lor  any  one  of  the  others,  so  that  it  is  en- 
dowed with  purchasing  power  e(|ual  to  theirs,  although  the 
purchasing  power  of  each  of  the  articles  is  now  one  fourth  of 
the  whole,  yet  as  it  is  one  fourth  of  a  whole  larger  by  one  third, 
as  in  the  former  instance,  therefore  the  ])urehasing  power  of  each 
article  continues  the  same  as  before  (cf.  Propositions  XLII.  and 
XLIII.),  and  so  the  total  amount  of  purchasing  power,  or  ex- 
change-value, in  this  world  has  increased. 

§  2.  There  is,  however,  a  form  hi  which  this  principle  may 
be  presented  which  seems  to  make  it  true  even  under  any 
changes  in  the  mass-ciuantities.'  This  form  is,  in  fact,  in  accord- 
ance with  the  way  in  which  tiie  principle  was  originally  hit  U})ou. 
Let  us  conceive  an  economic  world  containing  two  articles,  A 
and  B,  ecpially  valuable  at  first.  If  A  comes  to  rise  in  ex- 
(;hange-value  compared  with  B,  we  know  that  B  must  fall  in 
exchange-value  in  A.  Therefore  there  cannot  be  a  rise  both  of 
A  and  of  B  in  exchange-value  each  in  the  other,  or  of  both  in  both 
together  ;  nor  contrariwise  a  fall  of  both  together.  Now  if  A 
becomes  twice  as  valuable  as  B,  B  becomes  half  as  valuable  as 
A,  and  from  having  A  ^  B  and  B  ===  A  at  the  first  period,  we 
have  at  the  second  A  ==  2B  and  B  ==-  |A,  How  are  we  to  con- 
ceive of  a  constancy  here  of  the  exchange-values  of  A  and  B 
together?  If  we  add  together  the  equal  exchange- values  of 
A  and  B,  each  taken  as  a  imit,  at  the  first  period,  we  get  the 
sum  1  -)-  1  =  2 ;  and  if  we  add  together  the  unequal  exchange- 
values  at  the  second  period,  to  wit,  2  for  A  and  ^  for  B,  we  get 
2  -|-  I  =  2J,  which  is  not  the  same  as  before.  Or  again,  A  has 
risen  by  100  per  cent,  in  B,  ai^d  B  has  fallen  by  50  per  cent,  in 
A,  that  is,  A  has  varied  by  -|-  100  per  cent,  and  B  by  —  50 
per  cent.,  and  if  we  add  these  we  get  -|-  50  per  cent.,  which 
does  not  indicate  constancy.  The  indication  of  c(mstancy  is  to 
be  got  in  another  way.      At  the  first  period  we  may  multiply 

•'' This  wider  position  cannot  he  argued  for  on  tlie  ground  that  the  exehange- 
values  of  all  things  could  change  only  relatively  to  something  else,  which  is  self- 
contradictory  ;  for  they  also  could  not  remain  constant,  in  this  way,  except  in 
relation  to  something  else,  which  is  equally  self-contradictory  (cf.  Chapt.  111. 
Sect.  II.  '!>'i  4  and  !t).  What  we  are  here  dealing  with  is  not  the  exchange-values 
of  all  things  as  a  whole  compared  with  soinetiiiug  else,  but  a  total  of  some  sort 
of  all  the  exchange-values  of  all  iliiiigs  within  a  whole. 


rilK    DocriMNH    DKSCIMI'.KD  441 

tilt'  excliau'^c- value  of  each  article  as  expressed  in  the  other,  and 
the  product  is  unity,  for  1  x  1  =  1  ;  and  at  the  second  j)eriod 
the  multiplication  of  the  exchange-value  of  each  article  as  ex- 
pressed in  the  other  still  yields  unity  as  its  product,  for 
2x^  =  1.  Yet  if  a  third  article  had  come  into  existence,  say 
with  the  exchange-value  of  B,  on  multiplying  together  the  ex- 
pressions for  its  exchange-value  in  A  and  A's  in  it,  for  its  in  B 
and  B's  in  it,  and,  as  before,  for  A's  in  B  and  B's  in  A,  as  fol- 
lows, |x2xlxlx2xj,  the  product  still  is  unity. 

These  considerations  lead  to  two  propositions,  which  are 
curious,  but  of  little  utility.  They  are :  The  product  of  the 
ratios  of  aU  particular  exchange-vahies  is  unity  (Proposition 
LI.) ;  and,  The  product  of  all  variations  of  particular  exchange- 
valuer  is  unity  (Proposition  LII.).  To  prove  these  propositions 
we  may  try  them  on  some  simple  case,  so  far  as  possible  in  uni- 
versal form.  Let  us  suppose  that  we  have  an  economic  world 
containing  three  classes,  such  that  we  have,  at  the  first  period; 
one  mass-unit  of  [A]  for  every  two  mass-units  of  [B]  and  for 
every  three  mass-units  of  [C] ,  and  that  the  mass-unit  of  [A]  is 
worth  6 J  times  the  mass-unit  of  [B]  and  c^  times  the  mass-unit 
of  [C] .  Then  all  the  ratios  of  exchange-value  between  these 
things  are  expressed  in  the  following  list : 

A=c=6jB   =    6jB  =   c^C^^  CjC  =  c^C, 

B-IB   .  i-Ac^c^'^C^c  '^C, 
6j  b^         6,  6, 

6j  6^         b^         6j 

C^^B.^BciAc^  IC  =  IC, 

Cj  Cj  Cj 

C  =c^  ^i  B  c  ^^  B  r^  ~  A  =^    IC  =  IC, 

Ci  c,  c, 

C^^iBcAb  =  ^  A^  IC^  IC. 

^1         '^'i  ^. 

And  now  in  multiplying  all   these  ratios  together  we  find   we 


442  THE    CONSERVATION    OF    EXCHANGE-VALUES 

luive    two   i,   to    luiiltiply  by  two      ,  three  (^^   to  multiply   by 

1 
1      .     c  .  ,6 

three       ,  six  r^  to  niulti])ly   by  six     ' ,  and  eight  1  to  multiply 

together,  the  j^roduct  of  all  which  is  unity.  Then  at  the 
second  period  if  we  suppose  A  to  be  worth  b„  times  B  and 
('2  times  C,  and  if  we  suppose  the  same  or  any  other  num- 
ber of  mass-units  of  these  classes,  or  even  any  other  classes 
with  any  other  ratios  of  exchange-value,  treating  them  in  the 
same  way,  we  should  get  the  same  result.  But  when  we  mul- 
tiply together  the  variations,  it  is  plain  that  we  have  variations 
only  of  the  same  numbers  of  mass-units  in  each  of  the  classes. 
Let  us  tlien  suppose  we  have  tit   both   periods  those  already 

supposed.     The  variation  of  A  relatively  to  each  B  is  ,"  and  of 

b  .  c., 

each  B  relatively  to  A   ,  \  of  A  relativelv  to  each  C    "  and  of 
''  b./  ^  c, 

G  be 

each  C  relatively  to  A  -  ,  and  of  each  B  relatively  to  each  C  ,  ^ 

b,c. 
and  of  each  C  relativelv  to  each  B  ,^^  ,   while   the  variation  of 

each  B  to  each  B  and  of  each  C  to  each  C  is  1.  Thus  the  full 
list  is  : 


Variations  of  A  —  , " ,     , " 
0,       b. 


^2      ^2 


B-  1  , 


B-1, 


6,  b^c,  b^c^,  ^£2 

b.;  /v",  '  b.fi^'  b.fi^' 

b^  6,(*,,  /j,c.,  6jC'., 

b./  b^]'  Kfi,'  b^c]' 


V2'  'W  0.,^        '        ' 

b.fi,  b.,c,  c, 

/  ^              2    1              2    1  1  1             1 

'^    —   I          >  7          >  >         ')         ^? 

kf^  b£,  C, 


THE    DOCTI.'INK    lUX  lill'.KD  443 

And  liere,  too,  it  is  perceived  tliat  we  have  only  even  numbers 
of  reciprocals,  so  that  the  multiplication  of"  all  these  variations 
bv  one  another  gives  the  product,  unity.  And  the  same  \\(»uld 
be  the  result  with  any  numbers  of  thinus,  with  anv  |)articular 
variations. 

These  Propositions,  then,  are  of  little  im|)ortan('c  theoretical 
or  practical ;  for  as  the  products  of  all  the  ratios  at  every  ]ieriod, 
and  of  all  the  variations  between  any  j)eriods,  ar(>  always  unity 
— always  the  same, — we  gain  nothing  by  eomj)ariug  them.'' 

But  there  is  more  in  the  Proposition  placed  at  the  head  of 
this  Chapter,  something  which  renders  it  truly  a  doctrine  of  the 
conservation  of  exchange-value  (of  a  certain  kind),  under  the 
proviso  that  the  classes  and  their  mass-quantities  are  the  same 
at  l)oth  ])eriods, — or  that  we  confine  onr  attention  to  those 
classes,  and  to  those  portions  of  them,  that  are  the  same  (or 
.similar)  at  both  the  periods  compared.  •  In  order  to  understand 
this  we  must  find  the  method  of  measuring,  under  the  proviso 
stated,  the  variation  of  the  exchange- value  of  every  other  thing 
under  consideration,  as  Avell  as  that  of  money,  and  not  only  of 
their  exchange-values  in  all  other  things,  but  also  of  their  ex- 
change-values in  a//  things. 

IT. 

§  1.  Under  the  condition  that  the  mass-quantities  are  the 
same  at  both  periods  (or  that  we  are  dealing  only  with  the  mass- 
quantities  that  are  common  to  l)oth  periods)  we  know  that 
Scrope's  method,  applied  to  the  constant  mass-quantities,  is  the 
proper  method  for  measuring  the  variation  of  the  price-level, 
and  the  inverse  of  this  is  the  proper  measurement  of  the  vari- 
ation of  the  exchange-value  of  money  in  all  other  things.  Let 
us  now  examine,  under  its  universal  aspect,  the  simplest  hypoth- 
esis jjossible,  namely  the  fiimiliar  one  of  a  world  with  money  and 
two  commodities.  We  have  generally  stated  it  in  the  following 
schema 

I  .T  A  @  n,     y  B  @  ;3i, 
II  J-  A  (2)  «2    y  B  @  /%. 

*  Cf.  a  similar  useless  position  in  Chapt.  V.  Sect.  I.  J  o. 


444     MKASrilKMKNT    oK    KXCll  AN(;E-VA  HE    IN    AIJ.    THINGS 

Tliis  is  sufficient  when  we  are  treating-  of  the  exchange- value  of 
money  in  all  other  things  ;  but  it  is  incomplete  when  we  are 
treating  of  the  exchange-value  of  money  in  all  things.  Then 
we  must  include  a  statement  of  money  itself,  the  quantity  of 
which  must,  in  accordance  with  the  hypothesis,  be  the  same  at 
both  periods,  and  the  price  of  its  money-unit  is,  of  course,  at 
both  periods  a  unit.  Hence,  with  the  volume  of  money  repre- 
sented by  V,  the  schema  becomes 

I  r  A  @  «!     y  B  @,  ,3i     V  M  @  1, 

II  J-  A  @  a,    ij  B  (5  /3,,   V  :m  @  1. 

Scrope's  formula  for  the  variation  of  the  exchange-value  of 
immey  in  all  other  things  is 

this  beino-  the  same  as 


^01  "2        \  «2 


in  which  n.,"  =  .r«.,  -f-  //,!,,  and  this  again  the  same  as 

Mo, 


Mo, 


1    /       n.,         ,  /9.A 


that  is,  as  the  arithmetic  average  of  the  exchange-value  vari- 
ations w'ith  the  weighting  of  the  second  period.  We  have 
already  learned,  in  Chapter  V.  Section  YII.,  to  turn  such  a 
formula  for  exchange-value  in  all  other  things  into  the  formula 
for  exchange-value  in  all  things.  This  is,  returning  to  the 
second  of  the  above  fonnuhe, 

in  which  a.,"  =  .ra,  -f  _i/^i.,  -f  rl.      This  reduces  to 

Mai~xa^-\-^fJ^+  v' 

whicii  is  Scro])e's  formula  for  the  variation  of  the  exchange- 
value  of  money  in  all  things  (iuchiding  itself). 


Willi    CONSIAN  r    MASSKS  445 

Besides  this  we  want  Scrope's  fonmihe  for  the  variation 
of  the  exchanj]^e-vahic  in  <ill  thiniis  Ixith  of  [A]  and  of  [B] . 
These  we  may  ohtain  by  sini])lv  iinitatiiiLi  what  we  iiave  done 
for  money.  In  obtainino-  the  formula  for  money  we  first  stated 
the  conditions  in  terms  of  money,  the  money-unit  bein^  taken 
as  the  miit  of  exchange-vahie  at  eaeli  period,  and  the  exchange- 
values  of  the  other  mass-units  bcinu'  stated  as  they  relate  to  the 
exchange-value  of  this  money-unit  at  each  point.  Therefore, 
seeking  tlie  formula  for  [A],  we  must  first  of  all  state  the  rela- 
tions of  the  exchange-value  of  the  other  mass-unit  and  of  the 
money-unit  to  the  exchange-value  of  the  mass-unit  of  [A], 
taken  as  unit  at  each  period.      The  schema  then  is 

I  X  XOi  1     V  ^('1  —     V  yiOi    ^ , 

II  ..■  Xfi    1       ti  \\(„    -^      r  M  (n,^. 

As  we  have  here  modelled  everything  uj)on  the  })revious  schema 
of  money,  merely  changing  the  units,  we  may  merely  co])y  the 
formula  for  the  variation  of  the  exchange-value  of  money  in  (dl 
things ;  and  so  we  have 

-^02 'h '^ 


which  reduces  to 


^a2  _  «2(»^l  J-  i//^l  +  '')  ^  ^2    ^^a2 

Aai  ~  a^{xo.,  +  i/fi.,  -I-  v)  ~  «,    J/„i" 
Then  for  the  case  of  [B]  we  have,  first  of  all,  the  schema 
I  X  A  @  ^     </  B  («  1     (■  M  @  i 
II  r  A  @  J     V  B  fe  1     V  M  (a;  ^; 

whence  the  formula, 


44()  rHK    CONSEKVATION    OF    EXCH  A  N(;  H-\' A  [.IH 


«             1 

^a2           /'i                    ;'i 

What  18  thus  shown  for  money  and  two  classes  of  commod- 
ities is  plainly  perceived  to  be  extendible  to  money  and  any 
number  of  classes.  With  any  number  of  classes,  for  any  class, 
say  [L] ,  whose  prices  are  /.^  and  /,,  we  shall  have 

L„x  ^i    Mai 

We  thus  have  a  convenient  method  for  converting  the  formula 
for  the  variation  of  the  exchange-value  of  money  in  all  things, 
under  the  given  proviso,  into  a  formula  for  the  variation  of  any 
one  class  (that  was  included  in  the  formula  for  money)  in  ex- 
change-value in  all  things.  We  only  have  to  multi])ly  the  ex- 
j)ression  for  the  variation  of  the  exchange-value  of  money  in 
all  things  by  the  price  variation  of  the  class  in  question.' 

§  2.  Having  obtained  these  formula?,  we  may  now  discover 
the  meaning  of  the  Proposition  at  the  opening  of  the  Cha])ter. 
We  have  seen,  by  a  particular  example,  which  is  sufficient  to 
prove  a  negative,  that  the  expressions  for  the  exchange-values 
of  all  things  (or  classes)  in  all  other  things  do  not  sum  np  to  the 
same  figure  at  each  period,  even  when  the  same  things  are  in 
existence  at  Ijoth  periods.  We  shall  now  iind  that,  given  this 
condition,  the  universal  expressions  for  the  exchange-values  of 
all  things,  or  classes,  in  a/J  things  do  sum  uj)  to  the  same  figure, 

*A  similar  result  cannot  be  obtained  with  exchange-value  in  all  other  things  ; 
for,  while  we  have 

3fo^  ~  xa2  +1/(^2' 
we  have 

^^2  ^  ajiyl^i  +'v )  -B02  _  /JgCo-'a,  +v) 

^01     «i(2//52+^')  £oi~  l^ii^fi^+v)' 

anil  there  is  no  such  reduction. — All  thiis  is  in  agreement  with  Proposition 
XXXni.:  and  it  suggests  a  still  broader  Proposition. 


TIIK    iXxriMNK     KXl'LAI.NKI)  447 

wlu'ii  treated  in  a  sjx'cial  maimer.  'I'liis  is  to  miiltiph  the  ex- 
pressions for  tlie  exchange-values  at  each  pei-iod  by  the  weights 
at  the  first  |)eri()(l.  These  exj)ressi<)ns  at  the  first  periixl  are  t<t 
be  taken  as  luiits,  so  that  the  sum  of  tliese  units  multijilied  by 
the  weights  of  the  first  period  is  thus  expressed,  .»•«,  -(-  //^3',  -|-  v. 
And  at  the  second  period,  eom|)ai-e(l  w  ith  the  first,  the  expres- 
sions for  the  exchange- va hie  in  a//  tilings  of  all  the  classes  are 
the  expressions  above  discovered  for  the  variations  of  the  ex- 
change-values of  all  the  classes  in  af/  things.  Hence  what  we 
want  to  prove  is 


The  first  half  reduces,  and  we  get 

J/, 


{xa,^  -f  i/,i.,  -\-  r)        ■  =  .,7/,  +  t,.i^  -\-  r, 


which   is  evident  when  we  restore  the  full  ex]iression  for      '*^. 

Q.  E.  D.  Thus  (if  (fill/  tuH)  periods  in  lohieh  the  mass-qtinnfifies 
are  const((nf  the  .^otms  of  the  e.rcli(inf/e-valiies  in  ail  tJun</s  of  all 
the  claxsex  (or  of  all  things)  multiplied  hi/  the  weight.^  (d  the  first 
period  are  the  same,  so  that  eomparison  of  them  gives  miifi/,  itidi- 
eating  constancy  (Pro[)osition  LIIL). 

What  is  thus  easily  jiroved,  may  be  varied   in   the  following- 
way.     Of  the  three  exj)ressions 

we  know  that  at  least  one  must  be  below,  if  one  is  above,  unity, 
and  reversely.  We  know  this  on  general  principles  already 
reviewed.^  It  also  follows  as  a  necessary  consequence  from 
what  has  just  been  proved.  But  we  may  also  gather  it  by  sim- 
ple inspection  of  the  expressions,  when  stated  in  full.  FiCt  us 
suppose  that  the  exchange-value  of  money  has  risen  and  that 

M  <, 

_^"  =  r  (r  being  >  1 ).       Plic  ]mmccs  of  [A]  and  [B]  nnist  now 

'Propositions  IX.,  XV.,  XVJ.,  XXVIII.,  XXIX. 


448  Till-:  CONSERVATION  OF   i:.\(  ii.\N(;i:-v.\i.i  K 

(Mther  have  both  fallen  iiiiilurnily,  or  at  least  one  of  them  must 
have  fallen.  If  they  have  both  fallen  unilbrmly,  then,  in  the 
full   expression,  r  bein<>'  a  positive  ((uantity  that  cannot  be  neu- 

lected,  —  and  '-^  must  each  be  <ir{>ater  than  /•  ;  ami  conseciuentlv 
«.,  /^., 

—  and  '^  nuist  eaeh  be  smaller  than  -  .      But  /•  multi|)lied   bv  a 

fijrure  smaller  than  -  oiv(>s  a  result  smaller  than  unitv.      Or  if 

>-  r  '^ 

one  of  the  prices  has  fallen  less  (or  remained  constant,  or  even 
has  risen),  then   the  other  must  have  fallen  by  still   more  than 

-  ,  and  at  least  the  expression  for  the  variation  of  the  exchanffe- 
r 

value  in  all  things  of  the  class  (and  everything  in  it)  having  this 
price  variation  must  be  below  unity.  The  same  result  Avill 
follow  if  we  suppose  the  exchange-value  of  money  to  have  fiillen, 
or  if  we  started  with  either  of  the  other  classes  ;  and  also  it  ^vill 
be  obtained  if  we  enlarge  the  supposition  to  include  any  number 
of  classes.  Always,  if  any  one  of  the  expressions  is  on  one  side 
of  unity,  at  least  one  of  them  must  be  on  the  other  side.  (Where- 
fore if  all  but  one  are  known  to  equal  unity,  that  one  also  must 
equal  unity,  in  obedience  to  Proposition  XX.). 

This  being  so,  it  admits  of  demonstration  that  the  percentage 
of  the  variation,  or  the  sum  of  the  percentages  of  the  variations, 
of  exchange-value  in  all  things,  above  unity,  multiplied  by  the 
weights  at  the  first  period,  is  equal  to  the  sum  of  the  percentages 
of  the  variations,  or  to  the  percentage  of  the  \'ariation,  of  ex- 
change-value in  all  things,  below  unity,  likewise  midtiplied  by 
the  weights  at  the  first  period.  We  do  not  know,  in  the  ex- 
amj)le  before  us,  whicih  of  the  above  three  expressions  is  larger, 
or  which  smaller,  than  unity.  There  are  six  possible  ])ermuta- 
tions,  and  we  may  assume  one  of  them  as  a  s})ecimen.  Let  us 
assume  that  money  has  risen  in  exchange-value  in  all  things, 
and  that  tlu;  two  commodity  classes  have  each  fallen  in  ex- 
change-value in  all   things.      Then  the  percentages  of  the  rise 

of  money  (in  hundredths)  is  ,"'  —  1,  and  the  percentages  of  the 


Till-:  doctimm:  kxi'laixkd  449 

falls  of  the  commodities  art'  1 =■'  •   ,  '''  aiid  1  —  '^  •  ~^-^  .    We 

«,     J/,;,  ^-i^     l/„, 

therefore  wish  to  ])rove  that 

And  this  is  easily  done  because  the  equation  to  he  proved  re- 
duces, as  before,  to 

(;iv/,  +  ijfi.,  +  v)    y"  =  .r«,  +  y-i^  +  r  , 

and  so  again  is  evident  Avhen  we  restore  the  full  expression  for 
the  variation  of  money.  And  if  we  suppose  any  of  the  five 
other  permutations,  we  shall  always  find  a  similar  demonstra- 
tion. And  what  is  thus  proved  of  money  and  two  classes,  may 
be  extended  to  money  and  any  number  of  classes,  or  to  cases 
without  money  ;  for,  although  the  demonstration  becomes  more 
complex  with  a  greater  number  of  classes,  it  may  be  carried  out 
in  the  same  manner. 

The  meaning  of  what  has  just  been  proved  may  be  expressed 
as  follows  :  The  same  mass-quantities  of  all  classes  existing  at  both 
periods,  the  sum  of  the  percenta(/ej^  of  the  variation  of  every  class 
in  exchange-value  in  all  things,  eacli  nndfiplicd  hi/  the  weight  of 
the  class  at  the  first  period  (the  percentages  of  the  rises  being  treated 
as  positive  quantities,  and  the  percentages  of  the  falls  as  negative 
quantities),  is  zero.  (Proposition  LIV.)  This  zero  means  that 
the  common  variation  of  all  things  together  in  exchange-value 
in  all  things  is  by  zero  per  cent.,  which  is  constancy. 

H(n'e  we  have  used  percentages  in  the  ordinary  way,  reckon- 
ing them  in  the  starting  points  of  the  variations — at  the  first 
period.  Hence  it  is  only  natural  that  importance  should  be  as- 
signed to  the  variations  according  to  the  importance  of  the  vari- 
ants at  the  first  period.  And  conceiving  the  projiortions  of  the 
variations  in  this  the  usual  way,  we  find  that,  when  tlie  mass- 
quantities  are  constant,  there  is  what  we  have  called  arithmetic 
equality  between  all  the  proj)ortions  of  the  rising  variations  of 
exchange-values  in  all  things  and  all  the  proportions  of  the  fall- 
ing variations  of  exchange-values  in  all  things,  the  weights  of 
29 


450 


TlIK    CONSKKVA  rioN    OF    EXCHAXGK-V  AJ.T  K 


tlie  (ilasscs  at  the  first  period  being  assigned  t^)  their  variations. 
This  is  precisely  what  we  shonld  expect  in  snch  cases,  beeanse 
of  Scrope's  f'ornnda  holding  here.  For  we  have  learnt  that 
Scro])e's  fornnila  is  the  same  as  the  formnla  fur  the  (iritJimetic 
average  of  variations  Avith  the  weighting  of  the  fird  period. 

AVe  may  also  expect  more,  since  we  know  more  abont  Scrope's 
formula.  We  may  suspect  that  there  also  is,  in  these  cases, 
what  we  have  called  iKtniionic  vqmditii  between  all  the  pro])or- 
tions  of  the  rising  variations  of  exchange- values  in  all  things 
and  all  the  proportions  of  the  falling  variations  of  exchange- 
values  in  all  things,  the  weights  assigned  to  the  variations  be- 
ing those  of  the  second  period.  In  other  words,  the  sums  of 
the  percentages  of  all  the  rising  and  of  all  the  falling  variations 
harmonically  reckoned,  that  is,  reckoned  in  the  finishing  j)oints, 
at  the  second  period,  multiplied  by  the  weights  at  the  second 
period,  will  be  equal  ;  or,  if  added  together  as  positives  and  neg- 
atives, will  yield  zero  as  the  total  sum.  This  may  be  quickly 
proved  by  taking  the  same  ])ernuitation  in  the  al)()Ae  example, 
and  stating  its  percentages  (in  hundredths)  harmonically.  We 
wish,  then,  to  prove  that 


Ma,  _       ] 


r 


1  - 


M„, 


C  =  .>'f/. ,  ■! 


M,n 


1 


...        J  L         «i    il/„i        J 

and  can  do  so,  because  this  reduces  to 


+  'Ai  1 


M„. 


i'\ 


\ ; 


('-;t)-".(::£-')"'.tt 


Ma, 


-    1 


and  to 


(.r«,  -f  //,5f,  -h  /•)  \1."  =  .ra,  +  .'//i  +  '', 


/I/. 


which   is  evident  when  we  restore  the  full   expression   for  ,, 

-''-'02 

I  the  inverse  of  vi-""  ).  Q.  K-  13.  And  again,  as  we  may  ex- 
pect, we  find  that,  with  t\\'o  things,  or  classes,  constant  in  mass, 
and  (,'(|ually  important  ovei'  both   the  periods  together,  there  is 


TIIK    T)(»(    riMM']     KXI'LAINKl)  451 

r/('()incfric  ctjiKififi/  I)ct\\(rii  the  pro])orti()ns  of  tlicii-  variations  in 
excliango-valuc  in  all  t]iiii<>s  (vi/,,  the  two  things  or  classes  them- 
selves),— that  is,  the  percentages  of  their  variations  in  snch  ex- 
change-value, geometrically  reckoned  (reckoned  in  o|»|>osite  di- 
rections) are  equal.  For,  supposing  these  articles  to  be  jiriced 
in  ideal  money,  [A]  rising  and  [B]  falling,  what  w(>  have  to 
prove  is 


.'/ 

'^1 

+ 

.r 

— 

1 

I'i., 

y 

«., 

-f 

.?' 

/^i 

.'/ 

'     1 

+ 

x 

}/ 

+ 

x 

a. 


1  - 


■'■fl,  +  !' 


but  this  reduces  to 

which  is  the  very  condition  supposed,  so  that  the  equation  is 
true  when  this  condition  is  satisfied.  Q.  E.  D.  We  should 
find  also,  by  trial,  that  in  ordinary  cases  there  is  approximation 
to  geometric  ecpiality,  or  e(piality  between  the  sums  of  the  geo- 
metric percentages  of  all  the  rising  variations  together  and  of  all 
the  falling  variations,  if  these  percentages  are  multiplied  by  the 
weights  over  hofJi  the  periods.  But  these  two  positions,  about 
harmonic  and  geometric  e(juality,  are  of  little  utility,  especially 
the  last ;  and  they  do  not  merit  being  stated  in  formal  projiositions. 

Thus  we  have  found  in  what  sense  the  Proposition  at  the 
opening  of  the  Cha|)ter  is  true.  It  is  true  of  genei'al  exchange- 
value  in  all  things  rightly  measured,  and  with  proper  allowance 
for  the  sizes  of  the  classes.  It  is  not  true  of  general  exchange- 
value  in  all  other  things.^  Therefore  we  should  add  to  it  the 
words  "  in  all  fJiinr/s.^^ 

§  3.  In  evidence  of  what  has  here  been  universally  proved, 
and  as  suggestive  of  other  relations,  a  particular  example  may 
be  oifered.     Suppose  we  have  these  states  of  things  : 

'  I'arsons  tried  to  (leinonstrate  it  of  this  kiiiil  of  general  exchaiige-valiie. 
Hence  the  necessity  of  his  faihire.     See  Notes  1  and  2  in  Section  I.  of  this  Chapter. 


402  THE    f'ONHEKVATION    OF    EXCIIAN(;E-VALUE 

I  10  A  ^:,  1     5  B  @  4    fi  C  (a,  3    4  D  (o;  7     7  E  (5,  2 , 
II  10  A  @  1     5  B  @  4    6  C  @  6     4  D  @  7     7  P:  @,  2 . 

Here  we  have  only  one  price  variation,  and  this  price  variation 
shows  that  the  class  [C]  has  risen  in  exchauge-valiie  in  all 
other  things  by  100  per  cent,  (reckoned  in  the  ordinary  way). 
Now  if  we  measure  the  variation  of  money  in  exchange-value  in 
all  other  things  by  means  of  Scrope's  formula,  we  have 

P,  ^  10  +  20  +  36  +  28  +  4  _  108  ^ 

p^       10  +  20  +  18  +  28  +  4       90  ' 

indicating  a  rise  of  20  per  cent.,  which  means  that  money  has 
fallen  in  exchange-value  in  all  other  things  by  16§  per  cent, 
(for  ^9^03  =  0.8333 ).  This  means  also  that  all  the  com- 
modities which  have  remained  unchanged  in  exchange-value 
relatively  to  money  have  also  fallen  by  lOf  per  cent,  in  ex- 
change-value in  all  those  things  in  which  money  has  fallen  by 
that  jjercentage  (that  is,  in  all  the  things  which  are  other  to 
money,  namely  in  all  commodities).  Now  let  us  remove  money 
from  consideration, — or  let  us  suppose  we  have  been  dealing 
only  with  ideal  money  (money  of  account).  As  it  has  not  en- 
tered into  the  calculation,  its  removal  causes  no  change.  But 
now  all  commodities  become  all  things.  xVnd  the  classes,  [A] , 
[B],  [D],  [E],  which  remained  unchanged  relatively  to  money 
and  to  one  another,  are  still  altered  in  exchange- value  in  all  com-* 
modities,  tliat  is,  now,  in  <ill  things,  just  as  they  were  before. 
Thus  these  classes  have  fallen  by  16f  per  cent,  in  exchange- 
value  in  all  things.  And  the  class  [C]  has  still  risen  by  100 
per  cent,  in  all  otiur  things  ;  but  as  it  has  risen  by  100  per  cent, 
above  things  that  have  fallen  by  1()|  per  cent,  in  exchange- 
value  in  all  things,  it  has  risen  from  1  to  2  x  0.83^  =  l.()0§,  or 
by  6G|  per  cent,  in  exchange-value  in  all  things  (not  counting 
money,  but  including  itself).  This  may  be  shown  again  by  sup- 
posing that  at  the  second  period  all  the  prices  were  16f  per 
cent,  lower  than  they  were  supj)osed  actually  to  be.  Then  we 
should  have 

P,  ^    8^    +    16f    +    30    +    23^    +    lit   ^    90  ^    -.     r.r. 

p  10  +  20  +  18  +  28'+  14  90  ' 


TIIK    IMKTRINK    EX  I 'LA  INK  I)  45.3 

indicating  constancy.  For  here,  money  reniainini^'  stable,  tlie 
price,  and  consequently  the  exchange-value  in  all  things  otlier 
to  money,  that  is,  in  all  commodities,*  of  each  of  the  classes 
has  fallen  by  !(>§  per  cent.,  except  th(;  ]irice  of  C,  which  has 
risen  by  66|  per  cent.,  showing  that  its  exchange-value  in  all 
commodities  has  risen  by  <i6|  per  cent.  The  same  results  could 
also  have  been  obtained  directly,  by  making  use  of  the  method 
above  ex|)loited.  For  here  we  could  immediately  take  A  as 
the  unit,  omitting  money  altogether,  and  should  have 

A^  ^  10  +  20+l'S4-2.S+  14  ^  ;>0  _ 

A^,       10 -}- 20 +  36-^28-}-  14      108        '"  ^  ' 

and  for  the  variations  in  exchange- value  in  all  things  of  B,  D, 

E,  we  should  have  this  multiplied  by  |^,  ^,  f ,  but  for  C  we  should 

C. 
have  it  multiplied  by  f,  Avitli  answer     ~  =  1.66|,  showing  the 

rise  of  [C]  in  exchange-value  in  all  things  to  be  by  66f  per  cent. 
Now  66f  is  four  times  16|.     But  the  weights  of  the  classes 

[A],  [B],  [D],  [E]  are  four  times  larger  than  the  weight  of 
the  class  [C]  at  the  first  period.  Thus  four  classes,  four  times 
larger  than  the  class  [C]  at  the  first  period,  have  together  fallen 
4  X  16f  =  6(3f  per  cent.,  arithmetically  equalling   the  rise  of 

[C]  by  6 Of  per  cent.,  in  exchange-value  in  all  things  (namely 
the  five  classes  of  commodities).^ 

§  4.  In  this  form  the  Proposition  may  be  of  some  utility. 
Thus  if  we  suppose  only  two  classes  equally  important  at  the 
first  period,  and  the  mass-units  of  each,  A  and  B,  are  equally 
valuable  at  the  first  period  ;  if,  the  mass-quantities  being  con- 
stant, the  exchange-value  of  A  rises  to  double  that  of  B,  we  see 

*  Also  in  all  things  (including  money);  for  otherwise  the  method  of  conversion 
described  in  g  1  of  this  Section  would  be  at  fault.  This  is  because  money,  here, 
is  constant  in  general  exchange-value  of  both  kinds.  It  is  plain  that  any  number 
of  classes  could  now  be  added  with  constant  prices  (or  in  the  preceding  case  with 
prices  rising  20  per  cent.),  without  affecting  any  of  the  relations  thus  far  used. 
See  also  Proposition  XXXIV. — But  this  indication  is  7iot  true  of  their  exchange- 
values  in  all  other  things  (other  to  them  singly).     See  Proposition  XXXIII. 

'  Again,  at  the  second  period  the  four  classes  together  are  twice  as  important  as 
[C].  Reckoned  luirmonically,  ?.  e.,  in  the  finishing  points,  the  percentage  of 
■  their  falls  is  20  per  cent.,  and  the  percentage  of  the  rise  of  [C]  is  40  per  cent. — 
just  twice  as  much. 


454  THE    CONSERVATION    OF    EXCHANCrE-VALUE 

that,  whereas  at  the  first  period  each  ckiss  had  half  the  total  ex- 
change-vahie  in  the  world,  at  the  second  ])eriod  [A]  has  two 
thirds  of  it  and  [B]  one  third  ;  wherefore,  althoutili  A  has  risen 
by  100  per  cent,  in  exchange-value  in  the  other  thing,  and  B 
has  fallen  bv  50  per  cent,  in  exchange-value  in  the  other  thing 
— simple  geometric  variations, — the  exchange- value  of  [A] ,  and 
consequently  of  A,  in  all  things  has  risen  from  |  to  f  by  8.'>^ 
per  cent.,  and  the  exchange- value  of  [B] ,  and  consequently  of 
B,  in  all  things  has  fallen  from  ^  to  J  by  .33|^  per  cent. — which 
are  simple  arithmetic  variations.''  And  again,  in  an  example 
like  the  one  just  used,  if,  [AJ  and  four  other  classes  being  each 
equally  important  at  the  first  period,  A  rises  by  100  percent,  in 
all  the  others,  the  exchange-value  of  [A]  (and  of  A)  in  all 
things  rises  from  i  to  |  by  QQ^  per  cent.,  while  the  exchange- 
value  of  each  of  the  others  in  all  things  falls  from  ^  to  ^,  or 
together  from  ^  to  |,  by  1G|  per  cent. 

We  can  sum  up  these  relations  in  the  following  general  state- 
ment : —  When  in  an  economic  tvorld  the  niass-qaantities  are  con- 
stant, the  variation  of  the  exchange-valne  in  all  things  of  any  class 
(and  of  any  individual  in  if)  is  the  variation  of  its  share  of  the 
constant  total  exchange-value  or  piirclKtsing  jxuver  in  this  tvorld 
(Proposition  LV.). 

And  from  this  principle  and  the  j)receding  examples  we  can 
form  the  following  universal  statements  about  the  relationship 

'•  Tims  here,  dealing  with  exchange-value  in  all  things,  we  have  the  equality 
in  the  sums  which  we  could  not  get  in  Sect.  I.  i'2  when  dealiug  with  exchange- 
value  in  all  otiier  things.  Take  another  simple  example.  Suppose  B  is  twice 
as  valuable  as  A  at  the  first  period,  and  at  the  second  A  twice  as  valuable  as 
B,  and  that  the  numbers  of  A's  and  B's  are  equal  and  constant.  Then  [A] 
from  having  i  of  the  total  exchange-value  in  the  world  has  come  to  have  3  of 
it,  and  [B]  from  having  §  has  come  to  have  J.  Tlius  arithmetically  reckoned, 
A  has  risen  by  100  per  cent.,  and  B  has  fallen  by  oO  per  cent.,  in  exchange- 
value  in  all  things;  l)ut  the  weight  of  [B]  is  twice  that  of  [A]  at  the  first 
period.  But  again,  the  weight  of  [A]  is  twice  that  of  [B]  at  the  second 
period  ;  and  now,  reckoned  harmonically  (in  the  ending  points),  A  has  risen  by 
50  percent,  and  B  has  fallen  by  100  percent.,  in  exchange-value  in  all  things. 
And  again,  the  weights  of  [A]  and  [B]  over  both  the  periods  together  are  equal  ; 
and  now,  reckoned  geometrically  (from  the  opposite  points),  the  percentages  of 
the  rise  and  fall  in  exchange-value  in  all  things  are  equal  ;  for  [A]  has  risen  by 
T)0  per  cent,  reckoned  at  the  finish  while  [B]  has  fallen  by  50  per  cent,  reckoned 
at  the  start,  or  reckoned  in  the  opposite  directions  [A]  and  [B]  have  risen  and 
fallen  each  by  100  per  cent.     This  illustrates  iJ  2  of  this  Section. 


iiii-:  iMx  riM.M-;   k\im>aini;i)  455 

hctwt'C'ii  the  two  kinds  of  i»;enoral  cxchaii*!;('-valiU'.  The  mass- 
([uaiititics   reinaiiiiiio;  constant,  if  the  weight  of  a  class  at  the 

fir.sf  period  is      the  sum  of  the  weiglits  of  all  the  other  classes, 

and  if  this  class  rises  by  p  per  cent,  (in  hundredths,  reckoned  in 
the  ordinary  way)  in  exchange-value  in  all  other  things,  it  rises 

from  -  to  ■  in  exchany-e- value  in  a//  thinj^s,  or  by 

/•  4-  1       '■  +  1  +  /> 

rp 
:. per  cent.  :   while  the  others,  collcctivelv,   fall  from 

r  -\-  ]  -\-  p  ^ 

r  ''  . 

V  to  V hi    exchange-value    in    a/f    thinji^s,    or    by 

r  -I-  1        /•  -I-  1  -I-  p  ^  ^  '  ^ 

P 

per  cent.,  and  if  they  do  not  vary  amony-st  themselves, 

they  each  fall  by  this  percentage,  or  otherwise  they  fall  by  this 
percentage  on  the  average  ;  so  that  the  sum  of  the  percentages 
of  their  falls  equals  the  percentage  of  the  rise  of  the  one  class. 
And  if  this  class  falls  by  p'  per  cent,  in   exchange-value  in  all 

other  things,   it  falls  from               to  — -,-^| j  ,   in   exchange- 
rs/ 
value  in  <dl  things,  or  by . y  per  cent.  ;  while  the  others, 

o    7  .      J.  _|_   ]^   —  p'  i  ■> 

r  r  , 

collectively,  rise  from  ,  to  ^  ,  in  exchane-e-value  in 

•  '  1  -I-  r       r  -\-\  —p'  ^ 

P' 

(ill  thiup-s,  or  by z ;  per  cent.,  and  if  they  do  not  varv 

'^  '  -^    r  -\-  1  —  p'  ^ 

amongst  themselves,  they  each  rise  by  this  percentage,  or  other- 
wise they  rise  by  this  })ercentage  on  the  average ;  so  that  the 
sum  of  the  percentages  of  their  rises  equals  the  percentage  of  the 
fall  of  the  one  class. 

Here  we  have  an  answer  for  one  of  our  early  j)roblems,  posed 
in  Chapter  II.  Section  IV.  §  2, — but  an  answer  aj^plieable 
only  to  the  cases  when  the  mass-quantities  are  the  same  at  both 
periods.  We  see  that  the  larger  are  the  other  classes  (the  larger 
the  /•),  the  smaller  is  their  variation,  in  accordance  with  Proposi- 
tion XIX.,  and  the  more  nearly  the  variation  of  the  one  class 
in    exchange-value    in    all   things    comes    to    its    variation    in 


4-)()  THE    COXSERVATION    OF    EXCHAXdE-A'xVLUE 

exchange-value  in  all  other  things,  according  to  Proposition 
XXIV. 

§  5.  We  can,  however,  very  much  simplify  the  statement  of 
the  relationship  between  the  two  kinds  of  general  exchange- 
value,  by  starting  at  the  second  period.  In  the  first  complex 
example  above  used  (in  §  3)  another  relationship  stared  us  in 
the  face,  although  we  have  not  paid  attention  to  it  yet.  This 
is  that  the  exchange-value  of  [C]  in  a/l  things  (commodities)  has 
varied  two  thirds  as  much  as  its  exchange-value  in  all  other 
things, — or  its  variation  is  one  third  less  in  the  former  than  in 
the  latter  ;  but  also  the  weight  of  this  class  at  the  second  period 
is  one  third  of  the  weights  of  all  the  classes, — or  the  weights  of 
all  the  other  classes  are  two  thirds  of  the  weio-hts  of  all  the 
classes.  This  relationship  is  universal,  as  is  proved  by  the  fol- 
hnving  equation  expressing  it  in  regard  to  money  (on  the  sup- 
position that  money  is  rising  in  exchange-value), 

r  +  xa^  -f  yfi^  -f  ^  _^ 

V  -^  x«3  +  y/9,  +  ^      xa^  -\-  yjS^  -j- 

a^^i  +  M  + _  .  V  +  xa^^  +  yfi^^  + ' 

•'««2  +  yi^2  + 

M'hich  is  easily  seen  to  be  true  (or  if  we  suppose  money  to  be 
fill  ling  in  exchange-value  and  reverse  the  terms  in  the  first  half, 
the  same  relationship  is  evident).     Therefore 

The  mass-quantities  all  remainin(j  constant,  the  variation  of  any 
class  in  exchange-value  in  all  things  relates  to  its  variation  in 
exchange-value  in  all  otJier  tilings  as  at  the  second  period  the 
iveights  of  edl  the  other  classes  relate  to  the  weights  of  all  the  classes 
(Proposition  LA^I.).^ 

"  This  is  the  Proposition  referred  to  under  Proposition  XXV.  Jt  shows  the 
importance  of  the  word  "equal"  in  that  Proposition.  Here  among  constant 
mass-quantities,  as  one  class  alone  varies  in  price,  its  size  varies,  and  so  therefore 
does  the  proportion  between  the  variations  of  its  two  kinds  of  general  exchange- 
value.  Thus  as  the  class  rises  in  price,  its  size  increases,  and  therefore  the  rise  of 
its  exchange-value  in  all  things  lags  farther  and  farther  behind  the  rise  of  its 
exchange-value  in  all  other  things.  An  ordinary  class  rising  slightly,  its 
exchange-value  in  all  things  rises  almost  as  much  as  its  exchange-value  in  all 
other  things.  But  it  is  conceivable  that  the  class  could  rise  so  higli  as  to  become, 
at  the  second  period,  equal  to  all  the  other  classes  together.  Then  its  exciiange- 
value  in  a/l  things  has  risen  only  half  as  high  as  its  exchauge-value  in  all  other 
things. 


THE    IKXTIUXE    EXPLAINED  457 

This  Proposition  applies  to  tlu'  variations  in  the  two  kinds  of 
i2:eneral  exchange-value  of  every  individual  thino-  in  the  class  in 
question,  when  by  its  "  ex  change- value  in  all  oilier  things"  Ave 
mean  its  exchange-value  in  all  things  outside  its  own  class. ^ 
But  if  bv  the  exchange-value  of  an  individual  thinji:  "  in  all 
oihcr  things  "  we  mean  absolutely  its  exchange- value  in  all  other 
things  (including  all  other  individuals  in  its  own  class)  beside 
itself,  the  Proposition  must  be  modified.  Its  two  variations 
will  now  relate  to  each  other  as  the  total  exchange-values  (or 
money-values)  of  all  other  things  beside  itself  and  the  total  ex- 
change-values (or  money-values)  of  all  things  relate  to  each  other 
at  the  second  period. 

III. 

§  1 .  The  mensuration  of  exchange- value  in  all  things  may  be 
extended  to  all  possible  cases — with  mass-quantities  also  vary- 
ing. It  is  only  necessary  to  insert  the  article  itself,  whose  ex- 
change-value in  all  things  is  being  measured,  among  the  things 
in  which  its  exchange-value  is  being  measured,  and  to  invert 
the  formulae  above  found  to  be  the  best  for  measuring  the  varia- 
tion of  the  general  level  of  prices.  Thus  for  money  the  uni- 
versal schema  is 


I  1-1  M  @  1     :r,  A  @  «i     y■^  B  @  /J^ 
II  r,  M  @  1     a-.,  A  @  «,,     //,  B  @  Z?., 


The  formula  for  the  method  using  double  weighting  is 

^^'^■'  =  ^1  +  -^'i^i  +  yji,  +  _  v.,  +  x.ya^a.^  +  y-z^f^A  +  _ 

3/„i       v.,  -f-  x./x.^  +  yA  + V,  +  xya^a.^  +  yyAl^-2  +  

Now  it  is  evident  that  this  formula,  in  relation-  with  the 
formula  for  the  same  method  of  measuring  the  variation  or 
constancy  of  the  exchange- value  of  money  in  all  other  things, 
may  violate  Propositions  XXIL,  XXVI.,  XXXII.  and 
XXXV.  This  is  merely  a  continuation  of  the  defect  dis- 
covered in  the  preceding  Chapter.      But  the  other  two  superior 

**  This  is  the  sense  in  whicth  the;  term  has  generally  been  used  in  tliese  pages. — 
Of" course,  the  mass-quantities  being  constant,  the  variation  in  exchange-value  in 
«//  things  is  the  same  of  the  individual  as  of  its  class. 


458     MEASURE^FENT    OF    EXCII AX(;i:-VALUE    IN    ALL    THINGS 

methods  are  free  from  this  defect ;  wherefore  in  dealing  with 
exchange-value  in  all  things  the  employment  of  one  of  them 
becomes  imperative.  And  the  one  to  be  preferred  is  Scrope's 
emended  method.     The  formula  for  this  method  is 

J4i      ^i\i\  -\-  a.yx^.v,  +  iiyi/,!/.,  +  

Again,  reckoned  in  the  mass-unit  of  [A] ,  the  schema  is 


II  r,  M  @  -     .r,  A  @  1     y.,  B  @  'J 


The  formula  is 


;[  5        


a 


^2^   «1 -^ 

-  ■</f,t'   4-  s^X-X^  +  —  VViVo  +  

-  y^\  ^2  +  «i  ^^-^v^^  +  i'^y If  1^2  + )  ,,  M 

-  Kyi\v^^  f^2^^v^2  +  pyyxy2  + ) 


And,  reckoned  in  the  mass-unit  of  [B],  the  schema  is 

Ii'iM@-^     ;riA@"j     2/iB@l     , 

II  ?v,  M  @  ^      a-„  A  @  ^'     V  B  @  1 • 

And  the  formula, 

«  gT  ^^',''2  +   T  ^-^^-'^   +    <'/l.V2  +    

-"a2  _  /^i il] 

■^ai  ~   1     /  ,    «2    /  ,       / 

/'2  "  /'2 

3-  yv^v^  -j-  «,  n/.iv.  +  ,^y.'/,//,  +  )       o      ,r 

_  Pi ^  ih  .  -''"■-' 

y  (^'-t'-^   +    "2  ^^■'■v'-2   +    .^2^^//l.'/2   +    )  '    '  "' 

'2 


Willi      VAKVINf;      MASSES  459 

AikI  so  (»ii  tlirouoli  all  the  classes.  These  results  are  the  same 
as  in  the  partial  cases,  with  constant  mass-quantities.  They 
would  he  the  same  if  we  used  the  formula  for  the  <i'eometric 
method,  or  the  fornuda  for  the  method  with  doui>le  weiuhtinsr 
just  rejected,  or  any  other  formula  whats(X'ver.  Thus  in  general, 
When  the  expression  for  the  varialion  of  the  e.veheinf/e-ra/ue  of 
money  in  all  fhliu/s  has  hven  ohtained,  the  e.rjtressio)i  for  the  rari- 
ation  of  the  e.ivhaitf/e-ro/ue  of  an//  other  c/cixfi  In  nil  things  mail 
be  ohtaineeJ  by  intilfip/yiuf/  the  former  ej-presf<io)i  by  the  e.rpression 
for  the  price  variutifm  of  this  eldss  (Proposition   L\'I1.).' 

>j  2.  Here  it  is  impossible  to  find  any  ('(piality  in  the  sums  of 
the  percentages  of  the  variations,  nudtiplied  by  the  weights  of 
the  first  period,  or  in  any  other  regular  manner.  For  if  what 
was  found  to  he  true  in  >;  2  of  the  |)receding  Section  were  assumed 
to  be  true  here,  with  regard  to  three  classes,  we  should  have  to 
be  able  to  show  that 

but  on  restoring  the  present  value  of      "' ,   we   find    that   this   is 

not  necessarily  true, — nor  would  it  necessarily  be  so  if  we  used 
either  of  the  other  methods  for  measuring  variations  in  exchange- 
value.  It  may  be  true  sometimes,  but  only  accidentally.  There- 
fore, as  Cairnes  said,  the  aggregate  amount  of  purchasing  power 
in  an  economic  world  may  increase  or  decrease. 

Such  increase  or  decrease  can  be  measured  in  the  followmg 
manner.  Take  from  all  the  classes  the  mass-(piantities  that  are 
('(mimon  to  l)oth  the  periods  compared.  As  these  are  collectively 
constant  in  exchange-value  in  all  things  (amongst  themselves), 
their  presence  may  be  ignored.  To  be  examined  are  the  ex- 
cessive mass-quantities  of  some  classes  at  the  first  {)eriod  and  of 
some  classes  at  the  second  period.  If  the  exchange-value  of 
money  has  i)een  found  to  be  constant,  all  that  is  needed  is  to  add 
up  the  money- values  of  the  excessive  mass-cpiantities  at  the  first 
period,  and   the  money-values  of  the  excessive  mass-(piantities 

^  As  before,  this  Proposition  cannot  I)e  stated  of  exeluinge-value  in  all  other 
tilings.     It  expands  Proposition  XXXIII. 


4()()     :\[EAsrr!EMEXT    OF    p:XCHANC^E-yALUE    IN    AEL    THINGS 

at  the  second  period  ;  and  the  bahniee  between  them  will  shoAv 
whether  there  is  gain  or  loss,  and  to  what  extent.  But  if  the 
exchange-value  of  money  has  been  found  to  have  varied,  the 
()])eration  must  be  corrected  by  reducing  the  total  sum  of  the 
money- values  at  the  one  or  the  other  period — say  at  the  second 
by  multiplying  the  sum  first  obtained  by  the  expression  for  the 
variation  of  the  exchange-value  of  money — and  then  striking 
the  balance. 

This  procedure  does  not  show  the  relative  increase  or  decrease. 
After  all,  then,  the  completest  method  is  simply  to  add  up  all 
the  money-values  of  the  classes  at  each  period,  and,  after  reducing 
the  sum  at  the  second  period  to  a  sum  in  money  with  constant 
exchange-value,  to  compare  the  two  sums  together. 

Here  we  must  be  careful  as  to  which  general  exchange-value 
of  money  it  is  whose  variation  is  to  be  corrected.  If  we  are 
measuring  merely  the  increase  or  decrease  in  the  aggregate  ex- 
change-value of  commodities,  having  no  interest  in  the  increase 
or  decrease  in  the  total  exchange- value  of  the  class  money,  Ave 
should  use  merely  the  variation  of  money  in  exchange-value  in 
all  other  things  (that  is,  in  all  things  other  to  it,  therefore  in  all 
commodities),  at  the  same  time  leaving  out  of  account  the  quan- 
tity of  money. 

After  such  a  measurement  has  been  made,  and  the  increase, 
decrease  or  constancy  of  the  aggregate  amount  of  exchange- 
value  in  all  things  in  an  economic  world  has  been  ascertained, 
it  must  not  be  supposed  that  any  information  has  been  acquired 
as  to  the  increase,  decrease  or  constancy  of  the  aggregate  amount 
of  use-value  or  cost-value  in  that  world  ;  and  whether  the 
aggregate  amount  of  esteem-value  is  thereby  determined  is  a 
debatable  subject.  The  truth  of  this  statement  may  be  seen  by 
considering  the  conditions  when  the  mass-quantities  are  all  ex- 
a(!tly  the  same  at  two  different  periods, — when  we  know  the 
aggregate  amount  of  exchange-value  in  all  things  is  constant. 
For  that  the  aggregate  amount  of  the  cost-values  of  these  con- 
stant mass-quantities — determined  by  the  aggregate  amount  of 
labor  their  production  has  cost — need  not  be  the  same  at  both 
periods,  is  evident.      E(jually  evident  is  it   that  the   aggregate 


WITH      VAI!VIN(;      MASSKS  4(U 

amount  of  their  use-values  may  not  be  tlie  same.  For  instance, 
if  tlic  two  [)erio(ls  are  summer  and  winter,  the  aj^jLi;reji,ate  of  the 
total  utilities  of  all  the  classes  might  be  very  different  at  the  two 
periods.  On  the  other  hand,  it  would  be  a  ditilieult  (juestion  to 
decide  whether  in  this  latter  case  (on  the  supposition  also  that 
the  numbers  of  the  j)e()ple  are  constant)  the  ag<rregate  amount 
of  the  final  utilities  of  all  the  classes,  and  hence  the  aggrej^ate 
amount  of  the  esteem-values  of  all  the  constant  goods,  are  con- 
stant or  not.  For  if  the  goods  were,  say,  mostly  adapted  to 
summer,  a  few  of  them  Mould  have  no  (actual)  esteem-value  in 
summer  and  many  no  (actual)  esteem-value  in  winter,  while 
many  would  have  moderate  esteem-value  in  summer  and  a  few 
excessive  esteem-value  in  winter.  Whether  these  would  exactly 
counterbalance  or  not,  is  out  of  the  province  of  this  work  to  pass 
an  ojnnion. 

It  is  proper,  however,  here  to  state  that  the  old  and  nmch 
debated  question  concerning  the  measurement  of  the  irea/tJi,  of  a 
country  is  to  l)e  decided  by  saying  that  such  measurement  is 
the  measurement  of  the  increase,  decrease  or  constancy  of  the 
aggregate  amount  of  exchange-value  (of  all  commodities  in  all 
commodities)  in  that  coimtry  relatively  to  the  numbers  of  its 
inhabitants. 

§  .3.  Both  because  the  aggregate  amount  of  the  exchange- 
values  of  all  things  in  all  things  is  not  constant,  and  because  the 
measurement  of  general  exchange-exchange  is  not  known  with 
absolute  definiteness,  when  the  mass-quantities  as  well  as  prices 
vary,  it  is  impossible  to  construct  such  convenient  statements 
for  the  comparison  of  the  variations  of  the  two  kinds  of  general 
exchange- value  as  were  above  obtained  for  the  cases  when  the 
mass-quantities  are  constimt.  But  by  means  of  the  geometric 
average  of  the  price  variations,  with  its  proper  weighting,  ^\•hich 
method  we  know  to  be  a})proximately  accurate  (except  only  in 
rare  and  extravagant  cases),  we  may  obtain  almost  equally  con- 
venient approximately  correct  formuho  also  for  these  more  usual 

and  complex  cases.      Let  the  class  [A]  be  -  the  size  of  all   the 

other  classes  of  commodities  over  hofli  the  })eriods  together,  and 


462     -MKAsrUKMEXT    OP^    EXCHAXaE-VALTE    IN    ALE    THIN(;s 

let  US  suppose  its  price  has  risen  by  p  ]>er  cent,  (in  hundredths), 
while  the  prices  of  all  other  commodities  have  remained  un- 
changed. Then  the  exchange- value  of  money  in  all  other  things, 
/.  e.,in  all  conunodities,  has  fallen  approximately  from  its  first  con- 

(lition,  represented  as  unitv,  to     ■        ,  l>v  1  —      , 

^  •  '  ^        \  +  p    •  I        1  +  yj 

per  cent.  Therefore  every  one  of  the  other  commodities,  re- 
maining unchanged  in  relation  to  money,  has  fallen  approxi- 
mately by  this  jjercentage  in  exchauge-value  in  all  other  things 
beside  money,  that  is,  in  all  commodities  (including  itself). 
But  the  class  [A],  while  rising  by  p  ])er  cent,  in  exchange-value 

in   all   other  things,  has  risen  bv  p   per  cent,  above     ./      

'        1  +J0 

of  its  former  state,  represented  as  unity,  in   exchange- value  in 

all  commodities  (including  itself,  and  excluding  monev),  or  to 

P  .  P 

r-^rYZi^ —  above  its  former  state,  bv  ,.  ^:t77==  —  1    i>er   cent., 

V   ^  +P  ,         '      ^       "^^^ 

approximately.      Reversely  if  [A]  has  fallen  by  j/  per  cent,  in 

price,  and  therefore  in  exchange-value  in  all  other  things,  money 

has  risen  ap])roximately  in  exchange-value  in  all   things  from 

/•-l/'  1  /•   :    l/  T 

1  to    ^  l)v     I  —  1    per  cent.  ;  and  the  ex- 

'1  —  p      '      '        I  —  p 

change-value  of  every  one  of  the  other  commodities  has  fallen 
approximately  by  this  percentage  in  exchange-value  in  all  com- 
modities (including  itself).      But  the  class  [A]  has  fallen  by  // 

per  cent,  below     .  /  of  its  former    state   in   exchauije- 

l  1        1  _y>' 

value  in  all  conunodities,  or  to  ,.  ,  .- .        =^-  beh )W  its  f( )rmer  state 

/  -t- ix  -.  . 

V  1  -  P 

n 

by  1  —  .    ,  -— i)er  cent.,  approximately.      Or  if  we  desire 

to  make  similar  measurements  with  money  itself  included  in  the 
standard,  supi)Osing  its  weight  over  both  the  periods  to  be  s 
times  that  of  [A] ,  we  merely  have  to  add  this  to  the  indices  of 
the  roots  in  the  preceding  expressions.  The  larger  r  is,  it  is 
plain  the   smaller  will    be  the  opposite  variation  of  money  and 


WITH      VAI!VIN(i      MASSES  46 'i 

the  other  commodities  in  ex('haiii!:e-vahie  in  all   tliino-s,  and  the 

nearer  will  the  rise  of  [A]  in  exelian<>o-value  in  all   things  be 

to  its  rise  in  exchaflge-value  in  all  other  tliintrs,  according  to 

Propositions  XIX.  and  XX  FN'.,  already  referred  to  in  similar 

cases  above  examined. 

Again,  if  all  prices  vary  equally  l)v  +  y>  or  by  —  ji  per  cent. 

(rising  in  the  former  case  and  falling  in  the  latter),  or  if  they  so 

vary  on   the  average  (geometrically  measured),  their   variation 

may  be  comprehensively  represented  as  a  variation  from   1  to 

1  ±  p-     Then,  while  the  exchange-value  of  money  in  all  othet' 

1  ,  — » 

things  has  inversely  varied  from  1  to ,  either  by      -          or 

1  ±z  p  "^     1    -|-  ^9 

by per  cent.,  its  exchange-value  has  varied  aj)proximately 

from  1  to  ^ r    (in  which  /■  represents  the  number  of  times 

(l±p)^^'-  _         ^ 

all  the  commodities  are  larger  in  total  exchange-value  over  both 
the  periods  together  than  money).  Therefore  a  commodity 
which  has  varied  in  price  by  ±p  per  cent,  (the  same  as  the 
average  of  all  the  price  variations)  has  varied  from  1  to  1  =bp 
in  money  that  has  varied  to  the  one  or  the  other  of  the  above  ex- 
pressions according  as  the  one  or  the  other  of  its  general  exchange- 
values  is  considered.  Thus  in  all  things  other  to  money  (/.  e., 
in  ((//  commodities,  including  itself^  this  commodity  has  varied 

irom  1  to    '   =  1  oi-  to  '   =  i    that   is,  it  has   remained 

l+P  1-yj  ^ 

constant, — and  so,  of  course,  also  in  exchange-value  in  all  other 
commodities;  but  not  so  in  exchange- value  in  all  other  tilings 
(other  to  itself,  or  to  its  own  class,  including  money),  or  in  all 
things  (including  money).      In  af/  things  it  has  varied  from  1  to 

)         -(   r   or  to  y,       -^    r      or  from   1  to  (l±p)7+i,   while 

in  all  other  things  it  has  varied  from  1  to  (1  ±7^)^ — approxi- 
mately. 

§  4.  In  thus  measuring  the  exchange-value  of  money  in  all 
things,  or  of  anything  else  in  all  things,  including  money,  we 
must   know  the  (juantity  of  money  that   has  been    employed  at 


4(54     MKASfKKMKNT    OF    KX(  HANI  J  E- VALUE    IX    ALL    THINCJS 

each  period.  Here  is  a  problem  of  a  peculiar  nature,  never  yet 
discussed.  The  mere  comparison  of  the  quantity  of  money  in 
a  countrv  w  ith  the  cpiantities  of  commodities  in  the  market  at 
anv  one  time  is  not  satisfactory,  as  money  is  permanently  in  the 
mark(^t,  while  other  ooods  merely  How  through  the  market. 
Nor  is  tiiere  any  propriety  in  the  comj)arison  of  the  quantity  of 
monev  in  the  country  M'ith  the  quantity  of  goods  (reckoned  in 
monev-values)  bought  and  sold  during  each  of  the  periods  com- 
pared, because  the  lengths  arbitrarily  chosen  for  these  periods 
will  affect  the  quantities  of  the  goods  but  not  the  (piantity  of 
monev.  Xor  again  between  the  quantities  of  exchanges  effected 
with  money  and  the  quantity  effected  without  money,  because 
this  comparison  is  affected  by  banking  expedients,  and  banking 
expedients,  although  they  make  certain  kinds  of  money  less 
needed  and  more  scarce,  yet  do  not  lessen  the  importance  of 
monev  as  a  class — or  if  the  quantity  of  money  so  decreased  be 
counted  for  less,  the  quantity  of  the  substitute,  material  or  im- 
material, provided  by  the  banking  expedients,  ought  to  be 
counted,  which  is  still  more  difficult  to  estimate.  The  follow- 
ing suggests  itself  as  a  possible  solution  of  the  question.  Money, 
Avhen  used  as  a  mere  medium  of  exchange,  is  desired  only  for 
purchasing  other  things,  hence  not  in  itself,  and  so  is  incom- 
parable with  commodities.  But  money  is  desired  in  itself  when 
it  is  wanted  for  paying  debts  or  the  interest  on  debts.  Thus 
tlu!  (piantity  of  debts  falling  due  within  a  certain  period  (but 
onlv  those  contracted  in  an  anterior  period)  along  with  the 
(piantitv  of  interest  due  on  these  and  on  unexpired  debts,  seems  to 
be  a  quantity  comjiarable  with  the  (piantities  of  commodities  con- 
sumed within  the  period.  Yet  there  is  defect  also  here.  One 
of  the  reasons  for  wishing  money  to  be  stable  in  exchange-value 
is  because  of  this  very  debt-paying  (piality  it  possesses.  But 
the  larger  the  amount  of  debt  in  a  country,  and  the  larger  the 
amount  of  weighting  consequently  assigned  to  the  class  money, 
the  smaller  will  be  the  calculated  variations  of  money's  exchange- 
value  in  all  things,  whenever  a  variation  takes  place  ;  so  that  the 
greater  the  quantity  of  debt  and  consecjuently  the  greater  the 
nicetv  desired  in  the  calculation,  the  duller  becomes  the  calculation. 


Willi     \".\l;^■|^(;     mas^is  4(!.") 

Perhaps,  hcnvover,  after  all,  we  do  not  want  a  standard  con- 
sisting of  all  things,  inelnding  money,  hnt  only  a  standard  of 
all  commodities.  This  is  the  eoinniodity-standard  proper. 
Money  is  something  sui  cfcnerix,  which  we  eoininonly  (contrast 
with  commodities.  It  is  used  as  a  measure  of  the  exchange- 
value  of  commodities.  Hence  we  want  it  to  be  stable  in  ex- 
change-value in  all  eonmiodities.  To  be  sure,  if  it  be  stable  in 
this  exchange-value,  it  is  also  stable  in  exchange-value  in  all 
things,  including  itself;  wherefore  no  room  is  left  for  dispute  in 
regard  to  the  supreme  desideratum.  But  wlien  it  varies  in  gen- 
eral exchange-value,  and  tiiereibre  differently  in  exchange-value 
in  all  things  and  in  exchange-value  in  all  eonmiodities,  the  lat- 
ter variation  being  slightly  greater,  the  commodity-standard  not 
only  is  the  more  convenient  because  of  the  greater  and  conse- 
quently more  apparent  variation  of  money  in  it,  but  also  it  seems 
to  be  the  standard  in  reference  to  which  any  correction  in  the 
payment  of  debts  ought  to  be  made.  Still,  this  is  a  fine  ques- 
tion, which  will  become  of  practical  importance  not  till  society 
at  large  shall  have  grown  much  more  sensitive  to  variations  in 
the  exchange-value  of  money  than  it  is  at  present.  For  the 
present,  then,  we  may  assume  the  conmiodity-standard  merely 
for  its  convenience. 

§  5.  Adopting  this  standard,  we  see  that  a  variation  in  the 
price  of  any  commodity  exactly  represents  the  variation  of  this 
commodity  in  exchange-value  in  all  commodities  only  if  money 
has  remained  stable  in  exchange-value  in  all  other  (and  in  all) 
things,  or,  which  is  the  same  thing,  only  if  the  general  level  of 
prices  has  remained  the  same.  Yet  people  almost  always  re- 
gard a  variation  of  the  price  of  anything  as  an  indication  of  it^ 
variation  sim])ly  in  exchange-value  in  general,  that  is,  they  treat 
money  as  if  it  were  stable  in  general  exchange-value — as  if  it 
were  a  good  practical  individual  standard.  But  when  it  is  per- 
ceived that  money  has  not  remained  stable  in  exchange-value, 
there  is  need  of  correcting  this  false  inference  from  the  mere 
price  variation  of  any  article.  The  correction  can  be  easily 
made  if  the  variation  of  the  general  level  of  prices  (inversely 
showing  the  variation  of  the  exchange-value  of  money  in  all 
30 


4()'i      MKASriiKMK.NT    OK     KXc  lIANCiK-VAJ.lK    IN     ALL    TIILNGS 

cinninoditios)  has  alnnidy  Ix'cn  measured.  Thus  if  the  i^cueral 
level  of  j>rices  is  tuuud  ti>  have  falleu  hy  10  per  eeut.,  wliieh 
fall  uieaus  that  .!H)  nionev-unit  is  now  worth  what  I.IH)  niouey- 
unit  fornierlv  was  worth,  it  is  evident  tiiat  a  eomniodity  whose 
price  has  also  fallen  hy  10  ))er  eent.,  so  that,  instead  of  being 
held  at  1.00,  a  certain  (juautity  of  it  is  sold  for  .i>0,  has  re- 
mained constant  in  exehant;c-\alne  (in  the  C(tnnnodity-standard). 
And  a  commodity  that  has  fallen  o  per  cent,  in  price,  so  that 
what  of  it  before  cost  1.00  is  now  u'ot  for  .1)5,  has  really  risen 
in  such  exchauo-e-valne  from  .!H)  to  .!»•"),  which  is  a  rise  by  ."i.oo 
per  cent.;  while  a  commodity  that  has  remained  constant  in 
price,  has  really  risen  in  this  exchanoe-value  from  .!»(»  to  1.00, 
which  is  a  rise  by  11.11  per  cent.  On  the  other  hand  a  com- 
modity that  has  fallen  "20  per  cent,  in  price  has  really  fallen  in 
this  exchange- value  from  .90  to  .80,  or  by  only  11.11  per  cent. 

Now  as  one  of  the  uses  we  make  of  actual  prices  is  to  indicate 
the  variations  of  commodities  sim]»ly  in  exchange-value,  and  as 
this  indication  is  false  except  in  one  given  case,  it  may  not  be 
out  of  the  way  to  call  the  corrected  indication  by  the  term  true- 
price.  Of  course  the  actual  ])rices  are  always  true  ])rices  ;  for 
they  are  always  true  in  the  only  indication  which  they  properly 
make,  namely  of  the  variation  of  the  commodity  in  its  particular 
exchange-value  in  money.  By  "  true-prices "  is  meant  some- 
thing which  is  true,  not  necessarily  in  the  first  and  |)roper  in- 
dication made  by  actual  ]n'ices,  but  in  the  secondary  indication 
to  which  ])rices  are  habitually  put. 

This  term,  then,  being  pennitted,  we  can  most  conveniently 
find  true-])rices  by  making  use  of  the  princii)lc  above  enunciated 
in  Proposition  LVII.  That  Proposition,  to  be  sure,  refers  only 
to  the  variation  of  the  exchange-value  of  money  in  all  things, 
and  shows  only  the  method  of  obtaining  therefrom  the  variation 
of  any  commodity  in  exchange-value  in  all  things.  But  when 
\\■^^  know  the  variation  of  the  exchange-value  of  money  in  all 
other  things,  wdiich  are  all  connnodities,  then  by  multiplying 
this  by  the  price  variation  of  any  commodity,  we  obtain  the 
connnodity's  variation  in  the  same  standard  of  all  things  other 
to   nionev,    namelv    all   connnodities.      Thus  the   expression    for 


Wiril     \'AI!NIN(i     MASSKS  4<J7 

the  variation   of  money  in   cxcliangc- value    in  all  cdinmoclities, 
according  to  Scrope's  method,  bcino- 


J£ 


0\  (I. 


y.i\.r..  +  ,ty!/,!/,  +  ry~r,  + 


the  expression,  according  to  the  .same  method  of  mea.snrement, 
for  the  variation  of  A  in  ex  change- value  in  all  coimnodities  (in- 

chiding  itself),  represented  by   j  "   ?  would  be 


'■ac-i  "-^  "i 


^     v/,v,+:;'v^//,//=+;,' A=.+ 


\  .      . —       .    . —  ,  .       «,     M, 


{o.yx,x.^  +  ?yy^}l■^  +  r-y\^i  + ) 


01 


and  so  on  through  all  the  list  of  commodities.  Usually,  how- 
ever, the  first  measurement  would  be  in  the  form  of  obtaining 
the  variation  of  the  general  lev^el  of  prices.  But  this  is  merely 
the  inverse  of  the  variation  of  the  exchange-value  of  money  in 
all  other  things  (/.  c,  in  all  connnodities).  Hence,  the  operation 
of  measuring  the  variation  of  the  general  level  of  prices  having 
already  been  performed,  the  formula  for  finding  the  variation  of 
any  class,  say  [A],  in  exchange-value  in  all  commodities  is 

Thus,  in  general  :  The  variation  of  the  exohangc-ralue  of  ani/ 
commodity  in  the  commodifif-dandard  is  found  hi/  multiplying  the 
inverse  of  the  expression  for  the  variation  of  the  general  level  of 
prices  hi/  the  expression  for  its  own  price  variation  (Proposition 
LVIII.). 

Or,  if  preferred,  percentages  alone  may  be  dealt  with.  Let 
the  percentage  (always  in  hundredths)  of  the  already  measured 
variation  of  the  general  level  of  ])rices  be  represent(Ml  by  rb  -, 


408  (iENEIJAL    KX(llAN<;K-VAI>rE 

according  as  it  is  a  rise  or  a  fall  (if  there  be  constancy,  -  =  ()) ; 
and  let  =b  p  similarly  represent  the  percentage  of  tiie  known  va- 
riation of  the  prioe  of  the  commodity  in  question.  Then,  the 
desired  percentage  of  the  variation  of  the  true-price  being  repre- 
sented by  jj„(, ,  the  formula  is 

_  1  ±7J-(1  ±.t) 

and  this  variation  is  a  rise  if  the  result  be  positive,  a  fall  if  the 
result  be  negative,  and  it  is  constancy  if  the  result  be  0.  Or  if 
the  symbols  be  used  to  represent  full  ))ercentages  (in  integers), 
the  formula  is 

I  100  ±p-  (100  ±  -)  1 
^'"'-^1  100  ±-  |- 

It  must  always  be  borne  in  mind  that  when  the  variation  of 
a  commodity  is  so  measured,  the  measurement  is  of  its  variation 
in  exchange-value,  not  in  all  things,  nor  in  all  other  things,  nor 
even  in  all  other  connnodities,  but  ///  a/ 1  coiiunodltiei^  (including 
itself).^  Of  course,  if  the  commodity  be  found  to  be  constant  in 
exchange- value  in  all  commodities,  and  money  also  has  been 
found  to  be  constant  in  exchange-value  in  all  other  things,  then 
the  commodity  is  known  to  be  constant  in  all  the  other  kinds  of 
general  exchange-value.  Otherwise,  to  find  its  variation  in  any 
of  the  other  kinds  of  general  exchange-value  would  require  a 
sej)aratc  and  special  measurement  similar  to  that  by  which  the 
variation  of  money  was  measured. 

IV. 

§  1.  When  engaged  in  measuring  the  constancy  or  \'ariation 
of  the  exchange-value  of  anything,  it  is  essential  that  we  should 
be  clear  as  to  the  standard  we  are  using;  or  else  there  is  danger 

-All  this  is  said  on  the  supposition  that  the  eoniinoility  in  (juestion  is  one  of 
the  things  whose  prices  were  taken  into  account  in  measuring  the  variation  of  the 
general  level  of  prices.  If  it  be  one  of  the  things  necessarily  left  out  of  that 
calculation,  its  variation  in  exchange-value  determined  as  above  is  its  variation  in 
exchange-value  in  all  other  commodities  (by  "all"  being  meant  all  the  other 
commodities  that  are  included  in  the  standard,  it  being  impossible  to  include 
absolutelv  all). 


XEED    OF    DISTINGUISHING    ITS    KINDS  469 

of  oui-  tailing  into  inconsistencies.    This  risk,  and  this  need,  may 
be  illustrated  by  the  following  example. 
Suppose  at  the  first  period 

A  c  B  =  C^D  =  E  =  F  =  , 

and  at  the  second 

A  =  f  B  =  iC  =  D  c^E  =  F  : , 

the  only  changes  being  in  relation  to  B  and  C.  And  suppose 
the  classes  [B]  and  [C]  are  equally  important  over  the  two 
periods  together.  Then  we  know  that  the  exchange-value  of  A 
has  not  varied.     But  at  the  second  period 

B  :  21C  =  UA  =  UD  r  1|E  :  UF  - 

If  B  had  risen  so  as  to  become  equivalent  to  1|  of  every  other 
thing,  we  should  say  it  had  risen  by  50  per  cent,  in  general  ex- 
change-value (of  some  sort)  ;  and  as  it  has  risen  by  this  amount 
in  all  the  other  things  beside  [A]  and  by  more  than  this  amount 
in  [A],  it  has  evidently  risen  in  general  exchange-value  (of 
.some  sort)  by  slightly  more  than  oO  per  cent.  Also  at  the 
second  period 

^  ^  tBc.|A.fD.|E  =  |F  =  ; 

and  for  a  similar  reason  we  perceive  that  C  has  fallen  in  general 
exchange-value  (of  same  sort)  by  slightly  more  than  33^  per 
cent.  Now  then  we  might  be  led  into  the  following  argument. 
While,  at  the  second  period,  A  is  exchanged  for  f  B,  it  is  ex- 
changed for  33^  per  cent,  less  of  an  article  which  has  risen 
slightly  more  than  50  per  cent.  Therefore  it  has  risen  somewhat 
in  general  exchange- value  ;  which  is  contrary  to  our  first  conclu- 
sion. And  while,  at  the  second  period,  A  is  exchanged  for  1|^  C, 
it  is  exchanged  for  50  per  cent,  more  of  an  article  which  has 
fallen  slightly  more  than  33|^  per  cent.  Therefore  it  has  fallen 
somewhat  in  general  exchange-value  ;  which  is  contrary  both  to 
the  first  conclusion  and  to  the  immediately  preceding.  These 
two  opposite  conclusions  cannot  be  reconciled  by  saying  that  they 
counterbalance  each  other  ;  for  they  would  do  so  only  if  it  were 
necessary  for  the  owners  of  [A]  always  to  exchange  it  in  equal 
portions  for    [B]    and  for    [C] .     But,  according  to  the  above 


470  (JENEKAI.    KXCHANCK-VAHE 

ivasoiiing,  if  ;iny  one  excliangcd  all  his  [A]  for  [B],  he  would 
at  the  second  period,  iJ:et  more  general  exchange- value  than  he 
got  at  the  first  ;  and  if  another  exchanged  all  his  [A]  for  [C], 
he  would  at  the  second  period  get  less  general  exchange-value 
than  he  got  at  the  first — which  happenings  are  contradictory 
(because  of  Proposition  VII.).  Now  the  above  reasoning  would 
be  perfect,  and  would  lead  us  into  a  dilemma,  or  aporia,  casting 
doubt  over  our  whole  subject,  if  the  general  exchange-values 
referred  to  in  all  the  cases  were  the  same.     But  they  are  not. 

§  2.  In  the  first  case  A  is  constant  in  general  exchange-value 
of  both  kinds — exchange-value  in  all  other  things  and  exchange- 
value  in  all  thing's.  In  the  second  case  B  rises  by  more  than 
50  per  cent,  in  a  general  exchange-value  which  Ls  exchange- 
value  in  all  the  other  things  beside  it,  including  C,  but  excluding 
itself.  In  the  third  case  C  falls  by  more  than  33^  per  cent,  in 
a  general  exchange-value  which  is  exchange-value  in  all  the 
other  things  beside  it,  including  B,  but  excluding  itself — and 
hence  different  from  the  exchange- value  in  othei-  things  in  which 
B  rose  by  more  than  50  per  cent. 

On  the  other  hand,  B  in  exchanging  for  2  J  C  while  exchanging 

for  li  of  all   the   other  things,  exchanges   for  2^  of  an  article 

fallen   in   exchange-value,  while  it    exchanges  for   1|  of  other 

articles  with  constant  exchange-value.      If  then  C  has  fallen  in 

general  exchange-value  of  some  sort  by  exactly  33|  per  cent., 

9  —  3 
2^  C  is  e(|uivalent  to   — -       =  1|  times  the  former  C,  and  there- 

fore  to  1}  A,  to  H  D,  etc.  And  C  in  exchanging fi)r  only  ^  B 
while  exchanging  for  |  of  the  other  articles,  exchanges  for  ^ 
of  an  article  risen  in  exchange-value,  while  it  exchanges  for  |  of 
other  articles  with  constant  exchange-value.  If  then  B  has 
risen  in  general  exchange-value  of  some  sort  by  exactly  50  per 

4-1-2 
cent.,  I  B  is  e((uivalent  to  — ^ —  =  |  of  the  former  B,  and  there- 
fore to  2  A,  to  I  D,  etc.  Thus  in  these  cases  everything  comes 
out  right,  provided  that  B  and  C  have  so  varied  in  some  general 
exchange- value,  aud  that  this  is  the  same  general  exchange- 
value. 


NKKi)  OF   i)isTrN<;risinN(;    I'i's   kinds  471 

But  B  and  C  ha\<'  .-^o  varunl,  and  in  general  excliange-valiio 
which  is  the  same  ;  for  they  luive  both  so  varied  in  general  ex- 
change-value in  af/  things  (including  themselves).  For  A  has 
remained  constant  in  exchange-value  in  all  things,  including  B 
and  C  Therefore,  according  to  Proposition  LVII.,  when  B 
has  risen  by  50  per  cent,  in  A,  it  has  risen  by  50  per  cent,  in 
exchange-value  in  ((//  things,  although  it  has  risen  by  slightly 
more  than  50  per  cent,  in  exchange- value  in  all  things  other  to 
it.  And  when  C  has  fallen  by  33^  per  cent,  in  A,  it  has  fallen 
by  33J  per  cent,  in  exchange-value  in  nil  things,  although  it  has 
fallen  by  slightly  more  than  33^  per  cent,  in  all  things  other  to  it. 

This  is  evident,  as  regards  the  standard  of  all  things,  also  if  we 
represent  the  state  of  things  at  the  second  j)eriod  in  these  ways, 

B  r  IB  =  2iC  =  UA  r  UD  :  liE  c , 


and 


C=lCc=      4B=^     |A:       fl)r^     fEr 


for  here  it  is  evident,  in  the  first  case,  that  for  B  to  rise  in  ex- 
change-value in  all  things  by  50  per  cent.,  it  mnst  rise  still  more 
in  exchange-value  in  at  least  one  of  the  others  in  order  to  counter- 
balance the  fact  that  it  does  not  rise  in  exchange-value  in  itself, 
and  in  the  second,  that  for  C  to  fall  in  exchange-value  in  all 
things  by  33^  per  cent.,  it  must  fall  still  more  in  exchange- 
value  in  at  least  one  of  the  other  things  in  order  to  counter- 
balance the  fact  that  it  does  not  fall  in  exchange-value  in  inself.' 
This  solution  of  the  difficulty  has  been  anticipated  in  Proposi- 
tion XXXIII.  It  is  also  illustrative  of  Propositions  XII.  and 
XIII.  For  from  the  variations  of  B  and  C  in  relation  to  A  we 
might  gather  that  B  has  become  2^  times  as  valuable  as  C,  and 
C  "I  as  valuable  as  11  This  is  true  of  their  exchange- values  in 
all  things,  and  in  all  other  things  beside  them  both.  But  it  is 
not  true  of  their  exchange-values  each  in  all  other  things,  since 
in  all  the  things  other  to  it  B  has  become  somewhat  more  than 
2^  times  as  valuable  as  C  has  beccmie  in  all  the  things  other  to 
it,  and  inversely. 

*  It  will  be  noticed  that  in  l)oth  these  eases  the  extra  variation  of  the  one 
other  class  (equally  important  over  both  the  periods  with  the  (dass  in  question)  is 
the  square  of  the  common  variation  of  all  the  other  classes.  This  relation  is 
universal.  Hereby  is  given  a  partial  (and  in  complex  cases  an  approximate) 
answer  to  a  prol)lem  proposed  in  Chapter  II.  Sect.  V.  '^  tl. 


CHAPTER  XIV. 

THE  UTILITY  OF  MEASURING  TPIE  VARIATIONS  IN  THE 
EXCHANGE-VALUE  OF  MONEY. 

I. 

§  1.  Knowledge  of  the  (^'onstancv  or  variation  of  the  ex- 
change-value of  money  is  useful  both  for  theoretical  and  for 
practical  purposes. 

For  theoretical  purposes  it  is  useful  in  many  scientific  en- 
quiries, which  lead  on  to  conclusions  of  great  practical  impor- 
tance. Thus,  for  example,  there  is  a  prevalent  opinion  that  a 
rise  or  fall  in  the  exchange-value  of  money  has  considerable  in- 
fluence on  industry  and  general  prosperity,  partly  deleterious 
and  partly  beneficial,  the  one  iu  some  ways,  the  other  in  others. 
This  opinion  is  to  some  extent  based  on  experience  in  flagrant 
instances  when  there  could  be  no  doubt  what  course  the  ex- 
change-value of  money  was  taking  ;  but  as  yet  it  has  mostly 
been  based  on  reasoning  a  priori.  For  it  cannot  be  scientifically 
investigated  until  variations  in  the  exchange-value  of  mcmey  are 
scientifically  determined.  Its  scientific  investigation,  however, 
is  of  the  greatest  moment ;  for  if  there  be  truth  in  the  doctrine 
that  the  deleterious  influences  are  greater  than  the  beneficial, 
and  more  so  of  arise  than  of  a  fall,  the  detailed  knowledge  of 
such  influences  may  lead  to  corrective  and  even  to  ju-eventive 
measures.  Attem})t  may  perhaps  be  made  to  attack  the  source  of 
the  evil  by  regulating  the  exchange-value  of  money — both  to 
prevent  the  insidious  changes  of  metallic  money  over  long 
stretches  of  years  and  the  sudden  changes  of  credit  money  dur- 
ing short  periods.  For  this  ])urp(»s('  also  knowledge  of  the 
causes  (»f"  the  \:iri;itiuii^  in  the  cxchaiigc-valuc  oi"  nioiicy  will  be 

472 


riiEoiMOTicAL   IM  i;i*(>si;s  473 

necessaiy.  But  tlie  prevalent  opinions  on  such  causes  can  like- 
wise be  scientifically  confirmed  or  refuted  only  after  more  scien- 
tifically accurate  measurements  of  the  variations  in  question 
have  been  instituted  than  any  yet  made.  On  the  whole,  it  is 
apparent  that,  as  observed  by  Dana  Horton,  the  theory  of  the 
multiple  standard  is  "  the  key  to  the  entire  theory  of  money."  * 
Of  the  corrective  and  preventive  measures  more  will  be  said 
pr(!sently. 

§  2.  Then  again  we  have  need  of  knowing  how  commodities 
themselves  vary  in  general  exchange- value  ;  for  we  cannot  well 
investigate  the  causes  and  consecjuences  of  their  variations  in 
such  exchange- value  imperfectly  measured.  To  be  sure,  we  can 
easily  investigate  the  causes  of  the  variations  of  one  commodity 
in  exchange- value  in  another,  as  such  variations  are  plain.  But, 
after  all,  to  know  the  causes  of  these  particular  variations  is  not 
so  important  as  to  know  the  causes  of  the  general  variations, 
which  latter  have  commonly  been  the  subject  really  had  in  mind 
when  economists  have  treated  of  the  causes  of  variations  in  ex- 
change-value— especially  when  specifying,  as  they  so  often  do, 
that  they  are  dealing  with  the  causes  of  variations  in  prices 
under  the  supposition  of  money  being  constant  in  general  ex- 
change-value. Reasoning  on  this  subject  needs  to  be  based 
upon  experience,  and  therefore  we  should  be  able  to  convert 
the  supposition  of  money  being  constant  into  reality  by  correct- 
ing its  deviations  in  the  instances  taken  from  experience.  OtherT 
wise  a  commodity  may  appear  to  have  risen  or  fallen  in  general 
exchange-value  because  its  price  has  risen  or  fallen,  although  its 
general  exchange-value  may  really  have  varied  in  the  opposite 
direction.  Money  being  habitually  used  as  a  measure  of  general 
exchange-value  notwithstanding  its  own  variableness  in  general 
exchange-value,  we  need  to  correct  the  results  obtained  from 
measurements  with  this  imperfect  instrument,  after  first  meas- 
uring this  instrument  itself.  For  after  finding  its  variations 
we  can  adapt  the  variations  of  the  prices  of  commodities  to 
variations  in  general  exchange-value,  in  a  manner  already  ex- 
plained.     We  may  thus  obtain  what  have  above  been  called  the 

1  Silirr  and  Gold,  2d  ed.,  p.  40. 


474  UTILITY    OF    M()XKTAI!Y     M  KASIRKM  KXTS 

tr\ie-prkeit  of  commodities,  namely  the  prices  tlicy  would  liave 
had,  liad  motiey  remained  stable  in  cxchanj^e-valne,  and  had  no 
other  changes  occnrred.^  This  term,  to  roj^eat,  is  not  inap])ro- 
priate,  because  in  spite  of  the  variations  of  money  we  do  con- 
tinue to  make  use  of  prices  for  measuring  variations  in  the  ex- 
change-values of  commodities,  not  only  in  money,  but  in  things 
in  general ;  but  it  is  only  these  adapted  prices  in  an  invariable 
imaginary  money  that  are  true  for  the  latter  purpose. 

An  example  in  i)ointmay  be  taken  from  a  subject  now  agitating 
])ublic  opinion.  In  considering  whether  the  present  tendency  of 
productive  bodies  in  the  same  line  of  business  to  combine  and 
thus  avoid  competition  is  beneficial  to  the  country  at  large,  or 
otherwise,  one  of  the  items  discussed  is  whether  the  so-called 
"  trdists  "  already  formed  have  raised  or  lowered  prices.  Now 
to  discuss  whether  they  have  raised  or  lowered  merely  actual 
prices  is  only  to  discuss  whether  they  have  raised  or  loAvered  the 
exchange-value  of  their  products  in  relation  to  money.  But 
actual  exj)erience  of  mere  changes  in  price  of  any  ])articular 
class  of  connnodities  shows  only  a  change  in  the  relation  between 
the  general  exchange-value  of  the  class  in  question  and  tlie  gen- 
eral exchange-value  of  money  ;  and  does  not  show  whether  the 
change  is  on  the  side  of  the  commodity  or  on  the  side  of  money, 
or  how  much  on  the  one  and  how  much  on  the  other — that  is,  it 
does  not  sliow  how  nnieli  the  cliange  is  due  to  the  efforts  of  the 
j)roducers  of  the  commodity  and  how  much  to  the  efforts  of  the 
producers  of  money.  This  can  be  shown — for  we  are  dealing 
with  variations,  not  in  cost-value  or  in  esteem-value,  but  in  ex- 
change-value— only  by  investigating  furtliei-  the  relation  of  both 
these  classes  to  all  commodities ;  which  can  be  done  very  labor- 
iously in  regard  to  one  of  the  tilings,  ])referal)ly  money,  and 
then  very  easily  in  regard  to  the  other.  And  after  doing  this 
we  are  not  so  much  interested    in  the   relative  aecomplishments 

-  This  second  proviso  is  necessary  because  the  variation  of  money  in  general 
exchange-value  may  have  had  some  influence,  to  be  noticed  later,  to  eliange 
relative  excliangc-values  from  what  they  would  be,  had  money  remained  stable. 
But  the  variation  liaving  taken  phiee,  and  exerted  its  intiiiencc,  the  tria-^jHces 
still  indicate  the  variations  of  the  commodities  in  gciieial  cxchniiuc-x  :iliic  under 
sucli  influences. 


THEOUKTK'AL    ITIM'OSKS  475 

of  the  producers  of  the  given  eoininodity  nnd  (if  the  pnxhicers 
of  the  metal  used  as  money,  as  in  the  relative  accomplishments 
of  the  producers  of  the  commodity  in  (piestion  and  the  producers 
of  all  other  commodities.  Hence  our  interest  is  really  in  the 
variations  of  the  commodity's  true-price;  for,  although  such 
variations  do  n(»t  show  the  commodity's  variations  in  cost-value 
or  in  esteem-value,  they  do  show  the  relation  between  its  varia- 
tions in  these  values  and  the  variations  in  them  of  all  com- 
modities,— about  which  relation  more  will  be  said  in  the  next 
Section.  Yet  this  aspect  of  the  question  is  mostly  forgotten,  and 
the  question  is  often  thought  to  be  settled  by  an  appeal  merely 
to  the  actual  variations  of  the  prices  of  the  given  commodity. 
Thus,  for  instance,  in  a  recently  published  work  is  t<^)  be  found 
the  following  passage:  "  The  price  of  cotton-seed  oil  has  fallen, 
along  with  the  economic  improvement  in  its  production  intro- 
duced by  the  trust.  In  1878  the  average  price  of  standard 
summer  yellow  oil  was  47.04  cents  per  gallon.  In  1883,  the 
year  before  the  organization  of  the  trust,  it  had  only  fallen  to 
47.08  cents  per  gallon.  In  1887,  four  years  after  the  organiza- 
tion of  the  trust,  it  had  fallen  to  38.83  cents  per  gallon.  In 
other  words,  during  these  four  years  the  price  of  cotton-seed  oil 
fell  more  than  eight  times  as  much  as  it  did  during  the  five 
years  before  the  trust  Mas  formed."^  Here  no  reference  is  made 
to  the  fact  that  after  1878  industrial  conditions  took  an  upward 
swing,  which  lasted  till  1883,  and  was  followed  by  depression. 
Now  if  the  measurement  of  the  course  of  the  exchange-value 
of  money  during  these  years  provided  by  the  Aldrich  Report 
M'ere  reliable,  the  true-prices  of  cotton-seed  oil,  calculated  for 
the  two  later  years,  the  price  at  the  first  year  being  taken  as  the 
base,  would  be  44.37  cents  per  gallon  in  1883  and  41.89  cents 
per  gallon  in  1887.  In  other  words,  the  fall  before  the  organi- 
zation of  the  trust  was  by  7.45  per  cent.,  and  the  fall  after  the 
organization  of  the  trust  was  by  5.52  per  cent.,  so  that  the  true 
fall,  instead  of  being  eight  times  greater  in  the  later  period,  was 
only  three  fourths  as  great.*     Unfortunately  the  index-numbers 

^  G.  Gunton,  Trusts  and  the  public,  1899,  p.  15. 

•*  On  p.  218  of  the  same  work  the  prices  of  petroleum  are  given  for  a  similar 
purpose  (the  Standard  Oil   Company  being  estal)lished  in  1871  and   the  trust  in 


47G  UTILITY  OF  MONETAi.'v   ^^KAs^l;^:^[EXTS 

of  tlie  Aldricli  Report  not  only  do  not  refer  only  to  prices  in 
New  York,  where  the  above  prices  were  reported,  but  also  were 
calculated  in  an  improper  manner.  Except  that  the  figures  of 
the  Aldrich  Rejjort  cannot  in  most  cases  be  far  from  the  truth, 
Ave  are  loft  in  the  dark  as  to  the  exact  movement  of  the  true- 
])rice  of  any  commodity  during  the  years  preceding  and  succeed- 
ing the  organization  of  its  trust ;  and  to  the  extent  of  this  ob- 
scurity all  argumentation  on  this  subject  is  confused  and  con- 
fu-<ing. 

In  view  of  the  habitual  inattention  and  neglect  with  which 
the  subject  of  true-prices  is  treated  not  only  by  the  people  at 
large  but  by  economists  of  repute,  it  is  somewhat  discouraging 
to  find,  as  recalled  by  Dana  Horton,  that  the  need  of  observing 
them  was  pointed  out  more  than  two  hundred  years  ago.  In 
1()72  Pufendorf  wrote  :  "When  the  price  of  any  one  and  the  same 
thing  is  said  to  vary,  it  must  be  carefully  distinguished  whether, 
properly  speaking,  the  value  of  the  thing  or  the  value  of  money 
has  varied."^  And  yet  perhaps  the  first,  and  perhaps  even  the 
last,  writer  who  has  attempted  to  make  a  scientific  investigation 
concerning  true-prices  is  Professor  Laspeyres,  who   wrote  on 

1880).  Some  of  these  prices,  in  cents  per  gallon,  are  liere  given,  followed  by  the 
true-prices  calculated  from  the  figures  in  the  Aldrich  ReiJort : 

1863  30.7  30.7 

1867  20.5  16.4 

1871  21.7  18.1 

1873  16.0  13.4 

1877  15.0  14.7 

1879  8.125  8.6 

1880  9.125  8.7 
1884  ,  8.25  8.4 
1889  7.125  7.7 
1891                                  (i.O  7.6 

The  inferences  to  be  drawn  from  the  latter  figures  are  somewhat  different  from 
tliose  from  the  former.  In  the  eight  years  of  open  competition  the  true-price 
fell  41.05  per  cent.,  at  the  average  rate  of  4.39  per  cent,  per  anmim;  in  the 
nine  years  of  preponderance  of  the  Company  it  fell  51.93  per  cent.,  at  the  average 
rate  of  4.75  per  cent,  per  annai/i;  and  in  the  eleven  years  of  quasi-monopoly 
under  the  trust  it  fell  12.64  per  cent.,  at  the  average  rate  of  1.09  per  cent,  per 
attmcm. 

^  De  jure  naturce  el  gentium,  lib.  \'.  cap.  I.  (5  16.  Pufendorf,  however, 
would  have  judged  the  variation  in  the  exchange-value  of  money  by  that  of 
farm  land.  Yet  the  multiple  standard  was  known  still  earlier,  as  we  shall  see 
presently. 


I •  II A( T I (  A  L    I •  r H I '( »s i;s  4  (  < 

this  subject  about  thirty  years  ago.''  He  deserves  credit  for  so 
doing,  although  his  work  was  vitiated  by  a  faulty  method  of 
calculating  the  variations  of  money.  Other  writers  have  but 
barely  noticed  the  subject." 

§  o.  The  practical  purposes  above  mentioned  are  sought  to 
be  effected  in  the  schemes  for  correcting  the  variations  of  money 
in  its  function  as  a  measure  of  exchange-value,  and,  as  far  as  is 
possible,  for  preventing  such   variations. 

To  begin  with  the  former.  It  has  been  proposed  that  the 
mensuration  of  the  exchange-value  of  money  should  serve  as  a 
guide  in  credit  operations  extending  over  at  least  six  months, 
or  a  year,  and  longer.  The  design  is  that,  by  agreement  be- 
tween the  parties  at  the  time  of  contracting,  debts  of  all  sorts 
should  be  repaid  in  the  same  amouDt  of  exchange- value  as  was 
borrowed  or  bargained  for,  and  therefore  the  sum  of  monev 
pledged  should  be  paid  witli  addition  or  deduction  according  to 
the  fall  or  rise  of  money  in  exchange-value.  For  example,  if 
between  the  time  of  contracting  and  the  time  of  solution  money 
is  found  to  have  depreciated  10  per  cent.,  a  person  owing  100 
money-units,  knowing  that  100  of  the  new  units  are  worth  only 
90   of  the   old   units,  and  that   10  of  the  old  are  now  worth 

10  X     *,,    =  11.11,  must  pay  back  111.11   money  units;  and 

if  money  has  appreciated,  instead,  by  10  per  cent.,  the  same 
debt  is  discharged  by  the  repayment  of  only  90.91  money-units, 
the  sum  due  for  interest  being  increased  or  diminished  in  the 
same  proportion. 

This  proposal,  as  is  well  known,  was  revived  by  Jevons,  after 
having  been  suggested  as  a  serious  scheme,  perhaps  first,  by 
Joseph  Lowe,  near  the  commencem^it  of  the  century  just 
elapsed,  and  soon  afterwards,  in  dependence  upon  Lowe,  by 
Poulett  Scrope.     It  has  been  recommended  by  several  other 

^  Especially  in  his  essay  Welche  Waaren  werden  iin  Vcrlaufeder  Zeiten  i Miner 
theurer  f—Statislischen  Studien  zur  Geschichte  der  Preisen,  in  the  Zeitschrift  fiir 
die  gesaiumte  Staatswissensehaft,  Tiihingen  1872,  Iste  Heft.  He  had  previously 
touched  upon  the  subject  in  B.  25. 

''  E.  g.,  Sauerbeck  in  B.  79,  p.  59;t.— A  casual  use  of  a  solitary  true-price  is  to 
be  found  in  the  works  of  D'Avenel,  B.  117,  p.  6,  B.  11^,  p.  4. 


47S  T'TILITY    OF    MONETARY    MEAsrRP:MHNTS 

writers,  and  recently  by  Professor  Laves  as  something  new.** 
Yet  hardly  new,  even  when  I>o\ve  wrote,  was  the  idea  of  it, 
which  has  been  before  the  eyes  of  every  jurist  for  nearly  three 
hundred  years.  For  in  the  great  work  of  Grotius  is  the  fol- 
lowing passage  :  '^  Concerning  money  we  must  know  that  it 
naturally  possesses  the  capacity  to  pay  debts,"  not  in  its  ma- 
terial alone,  nor  in  its  denomination,  but  in  a  wider  respect, 
namely,  as  it  is  compared  either  with  all,  or  with  the  most 
necessary,  things  ;  which  estimation,  unless  otherwise  agreed 
u])(»n,  is  to  be  made  at  the  time  and  place  of  solution."  ^" 

§  4.  The  other  proposal  is  that  the  mensuration  of  the  ex- 
change-value of  money  should  be  employed  as  a  guide  for 
regulating  the  currency.  A  variation  of  money  being  detected 
before  it  has  had  time  to  go  far,  it  has  been  suggested  that  the 
government  can  restore  to  money  its  former  exchange- value  by 
increasing  or  decreasing  its  quantity,  and  by  performing  this 
operation  constantly,  it  can  keep  money  always  within  very 
slight  and  inconsiderable  deviations  from  a  permanent  ex- 
change-value— as  the  helmsman  steers  his  boat  by  arresting  its 

"  ]>owt',  B.  8,  pp.  278-279,  281-291 ;  Scrope,  B.  9,  pp.  407-408  (followed  by  R. 
H.  Walsh,  B.  13,  and  reviewed  by  Maclaren,  B.  17);  Jevons,  Money  and  the 
nice h(in ism  of  exchange,  pp.  328-333;  Horton,  Silver  and  gold,  pp.  30-43  ;  F.  A. 
Walker  (confining  it  to  persons  not  in  business),  Money,  pp.  161-103,  3Ioney  in 
its  relation  to  trade  and  industry,  pp.  70-77  ;  ^larshall,  in  a  paper  read  before 
the  Industrial  Kemuneration  Conference,  pp.  185-186  of  the  Keport,  London 
1885,  and  in  B.  83,  pp.  363-36.3  ;  T.  Laves,  Die  "  Warenwithmng  "  als  Ergdnzung 
der  Edclmetalwilhrung,  Schmollers  Jahrbuch  fiir  Gesetzgebung,  Verwaltung  und 
Volkswirthsohaft,  Leipzig  1890,  pp.  837-840.  The  scheme  is  entertained  by 
Laughlin,  History  of  bimetallism,  1885,  pp.  XI. -XII.,  and  Elements  of  polit- 
ical economy,  1887,  pp.  76-77,  and  by  (iitfen.  Recent  changes  in  prices  and  incomes 
compared,  .lournal  of  the  Statistical  Society,  London  1888,  pp.  54-55;  and  is 
recommended  as  a  substitute,  in  case  of  fiiilure  to  establish  the  next  scheme,  by 
II.  Winn,  The  multiple  standard,  American  Magazine  of  Civics,  Dec.  1895,  p. 
584,  and  Parsons,  B.  136,  p.  333.  The  present  scheme  was  also  favored  by  Zucker- 
kandl  in  B.  115,  pp.  249-250  ;  but  later  he  has  found  fault  with  it  for  not  allowing 
for  stal)ility  of  money  in  cost-value  (or  esteem-value)  when  its  exchange-value 
is  rising  because  of  improved  production  of  commodities  [and  not  of  the  money- 
material],  B.  116,  pp.  249-252. 

'■' Functio — fungibleness,  theal)ility  of  one  jjortion  of  money  to  i)c  paid  liack 
in  return  for  another  portion  of  money  borrow(!d. 

^"  J)e  jure  belli  el  pads,  1625,  lit).  II.  cap.  XII.  \  17.  The  (nnphasis  in  this 
passage  lies  upon  the  word  "  naturally."  .Vccording  to  Grotius  the  multiiile 
standard  is  this  natural  standard  for  paying  debts,  although  it  has  never  been 
ciriployi'd  within  aTiy  State. 


im;.\(  ri(  Ai.  I'lnrosKs  479 

incipiont  deviations  fVoiii  the  true  course.  The  money  whose 
quantity  is  to  be  reguhited  has  generally  been  chosen  to  be 
paper  money,  issued  by  the  government  either  directly  or 
through  the  mediation  of  a  bank  or  banks,  and  either  incon- 
vertible or  convertible  into  a  variable  amount  of  metallic 
money  or  bullion.  In  the  latter  case  this  scheme  is  somewhat 
like  the  preceding,  except  that,  the  medium  of  exchange  down 
to  the  smallest  subsidiary  coins  being  co-ordinate  with  the 
largest  bills,  this  scheme  will  extend  to  all  payments  even  of 
the  smallest  and  shortest  contracts,  while  that  scheme  was 
recommended  only  for  large  and  long  contracts ;  and,  again, 
this  scheme  must  from  the  beginning  be  compulsory,  the  paper 
money  being  legal  tender,  while  in  that  scheme,  the  idea  was 
that  the  contraction  of  ol)ligations  payable  according  to  the 
multiple  standard  should,  at  least  until  the  practice  became 
customary,  be  voluntary.  In  the  present  scheme  the  sugges- 
tion has  sometimes  been  made  that  the  quantity  of  the  out- 
standing money  may  be  regulated  by  raising  or  lowering  the 
rate  of  discount  according  as  prices  rise  or  fall  ;  otherwise  the 
regulation  of  the  quantity,  by  some  other  method  of  extending 
and  contracting  the  issues,  would  be  anterior,  itself  influencing 
the  prevalent  rate  of  discount.  Such  a  scheme,  more  or  less 
definitely  worked  out,  has  been  frequently  recommended,  more 
or  less  vigorously."  It  has  even  been  extended  by  Professor 
Walras  to  metallic  money  in  one  of  its  species,  namely,  to  silver 
coins,  in  a  system,  as  he  calls  it,  of  bl/fon  rnjulatviir,  for  which 

^^  It  was  hinted  at  by  R.  Walsh,  A  Letter  to  Alexander  Baring,  Esq.,  on  the 
jyresenl  state  of  the  currency  of  Great  Britain,  in  the  American  Review  of  History 
and  Polities,  Vol.  II.,  1811,  pp.  275-277,  and  by  Scrope,  B.  !»,  pp.  418-419;  and  is 
said  to  have  been  recommended  by  W.  Cross,  ^Standard  pound  versus  pound 
sterling,  185(5.  A  very  imperfect  form  of  it  was  suggested  l)y  .levons,  op.  cit., 
pp.  327-328.  It  has  been  advocated  more  seriously  by  J.  Barr  Robertson  before 
the  Gold  and  Silver  Commission,  Second  Report,  188S,  qq.  G2il4-(J304 ;  A.  Wil- 
liams, A  'fixed  value  of  bullion'  standard. — A  proposal  for  preventing  general 
fluctuations  of  trade.  Economic  Journal,  .lune  1892,  pp.  28U-289 ;  J.  Conrad,  in 
Wissenschaftliche  Gutachten  iiber  die  Wahrungsfrage,  Berlin  1893,  pp.  33-34; 
O.  J.  Frost,  The  question  of  a  standard  of  value,  Denver  1894,  p.  2(1;  Osborne, 
op.  cit.,  p.  332;  Fonda,  B.  127,  pp.  158-195  ;  H.  Winn,  op.  cit.,  pp.  579-5S9 ;  ,1. 
A.Smith,  B.  129,  pp.  33-42;  Whitelaw,  B.  130,  pp.  20-22,  28-82;  Parsons,  B. 
13t),  pp.  1(12  ff.;  T.  E.  Will,  Stable  money,  Journal  of  Political  Economy,  Chicago 
Dee.  1S9S,  pp.  8.5-92. 


4S()  iriLlTV    OF    MoNKTAItV    MKAST  REM  KNTS 

the  existence  of  "  limping  bimetallism  "  in  many  countries  offers 
opportunity. '- 

§  5.  A  warning  should  be  given  to  the  advocates  of  either  of 
these  schemes.  This  is  that  there  is  no  such  thing  as  a  single 
variation  in  the  exchange- value  of  money,  or  of  gold  or  silver, 
the  same  throughout  the  whole  world,  or  between  adjoining 
countries,  especially  if  separated  by  shifting  tariff'  barriers,  or 
even  within  the  borders  of  any  one  fairly  large  country.  For 
the  changes  in  the  charges  of  transportation  and  of  intercourse 
cause  the  variation  in  the  exchange-value  of  the  same  mone- 
tary system  to  be  different  in  diflFerent  regions.  In  our  coun- 
try, for  instance,  the  variations  in  the  exchange-value  of  money 
at  New  York,  at  New  Orleans,  at  Chicago,  and  at  San  Fran- 
cisco, would  form  four  appreciably  different  series.  Each  of 
these  series  should  be  measured  by  itself,  no  prices  in  one  part 
of  the  country  being  mixed  up  with  prices  in  another  part. 
Then,  in  the  former  voluntary  scheme,  the  contractants  could 
use  the  index-numbers  of  their  own  locality.  But  in  the  second 
compulsory  scheme,  as  no  one  center  alone  should  be  favored, 
the  standard  for  the  whole  country  should  be  an  average  of 
the  variations  at  the  different  centers,  each  being  weighted 
according  to,  not  so  much  the  population,  as  the  total  wealth, 
of  its  region. 

On  the  other  hand,  to  the  opponents  of  any  such  schemes 
may  be  given  an  admonition.  There  is  a  not  uncommon  opin- 
ion, to  be  found  even  in  the  works  of  very  respectable  econo- 
mists, that  the  exchange-value  of  metallic  money,  and  among 
English-speaking  peoples,  of  gold  money,  is  normally  stable  at 
certain  levels,  and  that  ])eriods  of  variation  are  only  transitions 
from  one  level  to  another.  Accordingly,  when  nations  have 
felt  themselves  suffering  from  monetary  appreciation  or  de- 
preciation, assertions  have  confidently  been  made  that  only  a 
little  patience  is  needed,  as  the  condition   is  transitory  and  will 

i^In  the  .Journal  des  Economi.ste.s,  Dec-.  1,S7G,  May  1881,  Oct.  1882;  B.  71,  p. 
II  ;  I?.  (i!t,  pp.  2-.'?,  12,  16-18  ;  B.  70,  pp.  411,441-449,  478-484;  B.  71,  pp.  143-144, 
1  IS'  i,-,i,  1(;2-U«,  484.— Walrus  is  followed  by  Andrews,  B.  107,  pp.  36-46,  B.  108, 
p.  1  U  ;  and  was  partially  anticipated  hy  Mannc(|uin,  /,«  iiwnnaie  et  le  double 
eta/on,  1874,  p.  ."){>. 


IMMNCII'LKS    OF    criMMlNCV     KKCILATION  4X1 

soon  be  followed  by  a  settled  condition  when  money  shall  have 
found  its  new  level  ;  and  therefore  to  do  anything  now  will- 
only  be  to  disturb  that  future  happy  state  soon  to  be  entered 
upon  naturally.  Nothing  could  be  more  false.  There  are  no 
periods  of  stability  in  the  exchange-value  of  metallic  money. 
No  measurement,  among  the  measurements  as  yet  made,  of  the 
course  of  the  exchange-value  of  money,  has  ever  shown  any- 
thing but  perpetual  movement.  To  be  sure,  it  sometimes  has 
happened  that  the  upward  and  downward  courses  are  short  and 
nearly  counterbalance  one  another  over  an  epoch  of  several 
years.  But  such  occasions  have  been  the  exception,  and  not 
the  rule.  The  so-called  industrial  cycles  of  eight  or  ten  years 
have  rarely  been  at  the  same  average  level  of  prices.  The 
rule  is  rather  that  the  fluctuations  are  uneven,  and  have  a  rising 
or  falling  tendency  for  many  years  at  a  time,  the  durations  of 
these  epochs  themselves  being  various,  so  that  anticipation  is 
impossible  for  more  than  a  year  or  two  in  advance,  and  then 
with  great  uncertainty.  Natural  money  has  no  permanent  level 
of  exchange-value  ;  nor  can  we  even  rationally  hope  for  such 
a  level.  Perhaps,  when  the  variations  in  exchange-value  both 
of  money  and  of  commodities,  the  latter  by  means  of  their 
true-prices,  have  been  better  investigated,  it  will  be  found  that 
the  precious  metals  are  among  the  products  of  human  industry 
the  most  variable  in  exchange-value. 

II. 

§  1.  The  principles  upon  which  rests  the  scheme  of  regu- 
lating the  exchange-value  of  money  by  exercising  artificial  con- 
trol over  its  issuance,  deserve  our  further  attention.  In  this 
work,  however,  we  are  not  concerned  with  the  investigation  of 
causes  ;  and  so  the  quantity  theory  of  the  cause  of  the  exchange- 
value  of  money,  which  is  the  foundation  for  that  scheme,  is  not 
to  be  discussed  here.  It  may  be  assumed  for  the  sake  of  argu- 
ment. To  be  examined  are  mere  relations  between  prices  and 
exchange- values. 

In  the  preceding  pages  the  question  has  been  investigated  as 
to  what  price  variations  ought  to  take  place  in  order  to  com- 
31 


482  I'TIMTV    OF    MONETARY     M  KASIR  E.MKNTS 

pensate  for  given  price  variatious,  the  mass-quantities  or  the 
money-vahies  of  the  classes  also  being  given.  Xow  in  the 
actual  work),  wlien  one  article  rises  a  certain  percentage  in 
price,  it  is  inij)ossible  by  any  nuinij)ulation  of  the  quantity  of 
money  to  effect  a  fall  of  another  article  alone,  or  of  all  other 
articles,  to  exactly  the  ])roper  extent  for  counterbalancing  the 
given  rise.  For  any  manipulation  of  the  quantity  of  money,  if 
it  affects  prices  at  all,  affects  not  only  the  price  of  the  other 
article,  but  the  prices  of  all  other  articles,  and  not  only  these, 
but  the  price  of  the  first  article  itself,  and  so  destroys  the 
datum.     We  have,  however,  the  following  principles. 

Commodities  change  in  price  relatively  to  one  another  accondnf/ 
<is  they  change  in  exchange-value  relatively  to  one  another,  which 
latter  relationship  is  hardly  affected  by  changes  in  the  quantity 
of  money.'  Bat  changes  in  relative  exchange-values  may  mani- 
fest themselves  in  prices  in  a)i  infinitude  of  ways.  For  instance, 
if  we  had  the  articles  [A]  and  [B]  in  certain  quantities,  A  and 
B,  each  priced  at  one  money-unit,  and  therefore  equivalent,  and 
later  they  change  relatively  to  each  other  so  that  A  =  H  B, 
this  could  be  (1)  by  A  rising  in  price  to  1.50  and  B  remaining 
at  1.00,  (2)  by  A  remaining  at  1.00  and  B  ftilling  to  .06f ,  (3) 
by  A  rising  to  1.20  and  B  falling  to  .80,  (4)  by  A  rising  to  2.00 
and  B  rising  to  1.33|,  (5)  by  A  falling  to  .75  and  B  falling  to 
.50,  or  in  any  of  numberless  variations  upon  the  last  three 
combinations.  Which  of  these  shall  he  the  one  actually  to  take 
place,  depends  upon  the  rehdionship  later  existing  between  [^] 
and  [i)']  together  {along  with  all  other  commodities)  on  the  one 
ha)  id  and  money  on  the  of  her. - 

^  Hardly,  because  there  is  some  iiifiueiicc  :  (1)  there  is  a  tendeney  in  prices  to 
round  numbers,  so  that  they  ciiange,  not  continuously,  but  discretely — "by 
hitches  and  starts,"  to  borrow  a  phrase  from  V.  Bowen,  American  political  econ- 
omy, p.  411  ;  (2)  noticeabh;  changes  in  thetiuantity  of  money,  differently  affecting 
the  prosperity  of  different  protlucers,  influence  some  articles  sooner  than  others  in 
what  A.  Del  Mar  has  felicitously  called  a  "  precession  of  prices,"  The  xcience  of 
money,  Jyondon  1885,  Cliapt.  VIII. 

^  Money  is  thus  contrasted  with  all  other  things,  not  because  of  its  uuitjue 
nature  already  referred  to,  but  merely  because  its  exchange-value  in  all  other 
things  is  under  consideration.  If  we  were  considering  the  variation  of  any  one 
commodity  class  in  general  exchange-value,  we  should  simihirly  iiave  to  contrast 
it  with  all  utlier  tilings. 


PHI^•C•lI•Ll•:s    OF    (  TIMIKNCV     IIKC  i  f  LA'I'K  ».\  483 

Here,  again,  in  our  conception  of  the  actual  variations  in  the 
exchauge-vahies  of  the  commodities,  we  are  no  more  concerned 
with  the  causes  of  these  variations  than  we  iiave  been  hitherto. 
[B]  may  fall  in  exchange-value  in  [A]  because  [B]  is  now 
produced  more  cheaply  and  abundantly  than  formerly,  while 
[A]  is  still  produced  with  the  same  difficulty  and  in  no  greater 
quantity,  or  because  [B]  is  still  produced  with  the  same  diffi- 
culty and  in  the  same  quantity,  while  [A]  is  produced  with 
greater  difficulty  and  in  lesser  quantity,  or  because  of  three 
more  typical  combinations  of  such  changes ;  or  again  because 
the  demand  for  [B]  has  diminished,  while  that  for  [A]  has 
remained  the  same,  or  because  the  demand  for  [B]  has  re- 
mained the  same  while  that  for  [A]  has  augmented,  or  because 
of  three  more  typical  combinations  of  such  changes — making 
five  in  each  kind,  which  may  act  singly  or  in  unison,  thus  pro- 
viding us  with  as  many  as  thirty  five  typical  combinations  of 
causative  changes.  But  these  causes,  so  far  as  operating  be- 
tween [A]  and  [B],  or  between  these  and  all  other  commodi- 
ties, even  if  perfectly  and  completely  known  to  us,  in  no  wise 
inform  us  what  are  to  be  the  changes  in  the  prices  of  [A]  and 
of.  [B],  or  explain  to  us  why  they  are  such  as  they  are,  except 
as  regards  their  relation  to  each  other  ;  for  these  changes  are  in 
relation  also  to  money,  and  depend  upon  a  similar  interaction 
of  causes  (again  in  thirty  five  typical  combinations)  operating 
between  [A]  and  [B]  and  all  other  commodities  on  the  one 
hand  and  money  on  the  other.  In  other  words,  without  the 
factors  affecting  money,  the  factors  affecting  commodities  can- 
not alone  determine  prices  ;  nor  the  former  alone  without  the 
latter.  The  two  sets  of  factors  cannot  operate  independently 
of  each  other.  Of  course  the  changes  in  the  causes  may  be  in 
only  one  of  the  two  sets,  the  other  remaining  untouched.  But 
in  whichever  set  it  be,  the  effect  upon  the  exchange- value  or 
exchange-values  on  the  one  side  is,  ipso  facto,  an  inverse  effect 
upon  the  exchange-values  or  exchange-value  on  the  other. 
For  the  exchange-value  of  money  is  affected  as  well  by  the 
causes  affecting  the  production  and  quantity  of  commodities  as, 
inversely,  by  the  causes  affecting  the  production  and  (piantity  of 


484  ITIMTV    OF    -MONETARY    .MEASUREMENT 

money  ;  and  the  level  of  prices  is  only  a  manifestation  of  the 
effect,  in  whichever  way  it  be  brought  about. '^  Therefore  the 
residence  of  the  changes  in  the  causes  on  the  one  side  or  on  the 
other  should  have  no  influence  upon  our  opinion  concerning 
the  variation  or  constancy  of  the  exchange-value  of  money. 
This  is  to  be  decided  only  by  a  measurement  of  the  actual  con- 
ditions,— most  conveniently  by  the  measurement  of  the  inverse 
variation  of  the  general  level  of  prices.  Changes  in  the  causes, 
moreover,  if  taking  place  on  both  sides  simultaneously,  and 
influencing  both  sides  in  the  same  direction,  may  neutralize 
each  other  and  so  leave  the  exchange-value  of  money  constant. 
The  exchange- value  of  money  will  be  altered  only  (1  and  2) 
by  the  existence  of  such  changes  on  either  side  alone,  either 
with  an  upward  or  with  a  downward  influence,  this  influence 
being  neither  aided  nor  impeded  by  an  opposite  or  by  a  similar 
influence  coming  from  changes  on  the  other  side  ;  or  (3)  by  the 
influences  of  such  changes  on  both  sides  in  opposite  directions, 
either  upwards  for  commodities  and  downwards  for  money  or 
downwards  for  commodities  and  upwards  for  money,  these 
influences  combining  to  increase  the  divergence;  or  (4  and  5) 
bv  an  excess  of  the  influence  of  such  chansjes  on  either  side 
alone  operating  in  either  direction  over  the  influence  of  such 
changes  operating  in  the  same  direction  on  the  other  side.  And 
of  course  what  is  here  said  of  the  general  exchange-value  of 
money  may  be  said  of  the  general  exchange-value  of  any  class 
of  things  over  against  all  other  things. 

§  2.  Now  what  everbody  wants  in  regard  to  all  commodities 
is  that  they  should  individually  and  collectively  become 
cheaper  in  cost-value  and  in  esteem-value.  Many  persons, 
then,  have  inclined  to  think  that  what  is  desirable  is  that  all 
commodities  should  become  individually  and  collectively  cheaper 
in  exchange-value  also.  Here  is  a  mistaken  inference  due  to 
confusion  of  thought,  itself  due  to  the  confusing  use  of  a  single 
term  '<  value  "  (and  of  the  allied  adjectivial  terms  "  dear  "  and 
"  cheap  ")  to  express  what  really  are  distinct  ideas,  the  genus 
being  used  for  the  species  because  the  species  have  not  been 

•'' See  abovf.  Chiipt.  11.  Sect.  IH.  ?  1. 


PRIXCrPLKS    OF    CURRENCY    RPXiUI.ATION  48") 

sufficiently  distinguished,  in  the  error  which  logicians  call   the 
fallacy  of  undistributed  middle.' 

The  inference  is  mistaken,  as  is  plain  from  the  fact  that  it 
violates  Proposition  LII.,  and  what  is  inferred  is  impossible. 
To  renew  the  proof  of  this,  let  us  suppose  that  the  alleged  de- 
sire for  the  cheapening  of  everything  in  exchange-value  is  sat- 
isfied by  the  successive  equal  cheapening  of  everything.  If 
[A]  becomes  cheaper  in  exchange- value  first,  ip-^o  jdc-io,  as  we 
know,  everything  else  becomes  slightly  dearer  in  exchange- 
value.  If  later  [B]  becomes  equally  cheaper  in  exchange- 
value,  again  everything  else  (including  [A] )  becomes  slightly 
dearer  in  exchange-value.  Also  [B]  is  then  as  cheap  as  [A] , 
wherefore  [A]  is  as  dear  as  [B] .  Again,  if  [C]  becomes 
equally  cheaper  in  exchange-value,  everything  else  (including 
[A]  and  [B]  )  becomes  slightly  dearer  in  exchange-value  ;  and 
now  [A] ,  [B]  and  [C]  are  equally  cheap  and  equally  dear. 
And  the  successive  fall  of  everything  else  will  tend  to  raise 
slightly  the  exchange-value  of  everything  else  (including  the 
things  already  fallen).  Every  single  thing,  therefore,  at 
some  time  falling  in  one  jump  the  full  extent  of  the  common 
fall  supposed,  and  then  effecting  a  slight  rise  in  all  the  others, 
will  itself  be  raised  in  very  small  stages  by  the  falls  of  all  the 
others,  whether  before  or  after,  or  partly  before  and  partly  after, 
its  own  fall.  Thus  when  all  but  one  have  fallen,  that  last  thing 
will  be  dearer  in  exchange-value  than  it  was  at  the  commence- 
ment by  the  inverse  of  the  fall  of  all  the  rest.  When  it  too 
falls,  the  last  rise  in  the  others  will  be  consummated,  and  as  the 
last  thing  has  fallen  back  to  equivalence  with  all  the  rest,  [A] 
and  [B]  and  [C]  and  [D]  and  the  rest  through  the  whole  list 
down  to  the  last  one  now  being  equivalent,  their  exchange- 
values  will  be  in  exactly  the  same  condition  as  they  were  at  the 
commencement,  that  is,  everything  will  have  the  same  exchange- 
value  as  before,  and  the  very  supposition  by  which  everything 
is  made  successively  to  fall  equally  in  exchange-value  shows 
that  when  the  operation  is  completed  nothing  is  fallen  in  ex- 

■*Notice  that  we  do  not  want  tilings  to  fall  in  use-value.  Such  a  fall  would 
mean  an  approach,  on  our  part,  to  insensibility.  Rather  we  would  prefer  that 
things  should  rise  in  use-value. 


486  UTILITY    OF    MnSKl'.WW     M  KAsr  HKMKNTS 

chauge-value.  Of  course  if  we  should  attempt  to  suppose  that 
all  things  fall  equally  in  exchange-value  together  and  at  once, 
the  same  result  would  be  involved,  which  means  that  nothing 
is  fallen,  although  all  are  supposed  to  have  fallen,  so  that  such 
a  supposition  is  a  contradiction  in  terms.  And  if  the  things  are 
all  supposed  to  fall  in  exchange-value  unequally,  or  if  only 
some  of  them  are  supposed  to  fall,  the  supposition  remains  true 
that  these  latter  are  fallen,  or,  in  the  former  case,  that  those 
which  were  supposed  to  fall  most,  still  are  fallen.  But  at  the 
same  time  it  is  true  that  the  things  supposed  to  fall  least  are 
risen  in  exchange-value.  It  is  impossible  to  carry  out  the  con- 
ception of  all  things  falling  together,  equally  or  unequally,  in 
exchange- value. 

The  reason  why  any  difficulty  should  appear  in  this  subject 
is  that  it  is  possible  for  people  to  make  the  supposition,  or  to 
entertain  the  thought,  that  some  or  all  things  have  fallen  in 
exchange- value,  and  at  the  same  time  to  forget,  or  perhaps 
never  to  perceive,  that  every  fall  they  suppose  in  exchange- 
value  involves  also  a  rise  in  exchange-value.  Of  course,  then, 
while  entertaining  the  one  thought,  and  excluding  the  other, 
they  may  think  of  all  things  falling  in  exchange-value.  But 
they  can  do  so  only  because  they  are  thinking  imperfectly. 

And,  to  repeat,  the  reason  why  people  have  thought  so  im- 
perfectly is  that  they  confound  exchange-value  with  cost-value 
or  with  esteem-value  ;  for  in  these  latter  kinds  of  exchange- 
value  it  is  ])ossible  for  everything  to  fall,  and  to  remain  fallen, 
the  fall  of  one  thing  in  such  values  not  having  any  necessary 
influence  to  raise  any  other  things  in  such  values.  A  passage 
which  may  serve  as  a  locus  clas,slGiis  of  this  confusion  of 
thought  is  the  following,  taken  from  the  Political  Economy  of 
Malthus.  In  it,  to  be  sure,  Malthus  uses  the  term  "  exchange- 
able value  "  ;  l)ut  it  must  be  remembered  that  he  had  no  terms 
whereby  to  distinguish  from  exchange-value  the  other  kinds  of 
value  except  use-value,  so  that  by  "  exchangeable  value,"  as  by 
"value"  itself,  he  variously  referred  to  all  the  kinds  of  value 
excei)t  the  last.  Now  after  supposing  a  case  of  equal  improve- 
ment in  the  production  of  all  goods,  he  [)roceeds  to  ask  and   to 


l'i;iX(  II'TJ'.S  OF  CURRENCY  REGULATION        487 

answer  as  follows  : — ''  Ctin  it  be  asserted  with  any  semblance  of 
correctness,  that  an  object  which  under  these  changes  would 
command  the  same  quantity  of  agricultural  and  manufactured 
products  of  the  same  kind,  and  each  in  the  same  proportion  as 
before,  would  be,  practically  considered  by  society  as  of  the 
same  exchangeable  v^alue  ?  On  the  supposition  here  made,  no 
person  would  hesitate  for  a  moment  to  say,  that  cottons  had 
fallen  in  value,  that  linen  had  fallen  in  value,  that  silks  had 
fallen  in  value,  that  cloth  had  fallen  in  value,  etc.,  and  it  would 
be  a  direct  contradiction  in  terms,  to  add  that  an  object  which 
would  purchase  only  the  same  quantity  of  all  these  articles,  which 
had  confessedly  fallen  in  value,  had  not  itself  fallen  in  value."  '* 
Here,  of  course,  if  by  "  value  "  be  meant  cost-value,  or  esteem- 
value,  the  conclusion  would  be  perfectly  correct.  And  here 
Malthus  himself  did  have  in  mind  a  mixture  of  cost-value  and 
esteem-value,'*  and  his  confusion  consisted  in  still  calling  it 
'^  exchangeable  value,"  which  nine-tenths  of  his  readers,  and 
himself  too  at  times,''  take  to  mean  what  the  terra  itself  prop- 
erly means.  His  conclusion,  then,  is  both  correct  and  incor- 
rect. For  the  reasoning  is  perfect  on  condition  that  the  falls 
of  all  the  separate  classes  of  things  are  independent  of  one 
another,  and  that  there  are  no  rises  really  involved,  but  sup- 
pressed. Such  is  the  case  with  the  falls  in  cost-value,  as  also 
in  esteem-value  (or  at  least  to  some  extent, — that  is,  the  condi- 
tion is  a  possible  one).  But  such  is  not  the  case  with  falls  in 
exchange-value.  \Yith  regard  to  this,  the  appearance  of  cor- 
rectness in  the  reasoning  is  acquired  only  by  suppressing  the 
counterbalancing  rises  necessarily  involved  in  every  fall,  wholly 
forgetting  or  ignoring  the  interdependence  and  correlation  of 
all  variations  in  this  kind  of  value. 

§  3.  The  desire,  then,  that  all  things  should  become  cheaper 
in  exchange-value  is  impossible,  absurd,  and  inane.     But  there 
is  another  confusion  of  thought  possible  in  this  connection,  and, , 
because  possible,  often   fallen  into,  which   is  not  so  empty  and 
harmless.      Variations   in    prices   are  variations    in   exchange- 

''Set'oiKl  ed.,  p.  "18. 

«  Cf.  ibid.,  p.  60. 

'Cf.  ibid.,  pp.  5(taiul  CI. 


488  LTILITY    OF    :srONETAl{Y    MEASl'IJEMENTS 

values  ;  and  therefore  variations  in  exchange-values  have  been 
confounded  with  variations  in  prices.  And  as  people  have  en- 
tertained the  contentless  desire  that  all  commodities  should  fall 
iu  exchange-value,  they  have  identified  this  with  the  desire 
that  all  commodities  should  fall  in  price.  Or  perhaps  this 
desire  has  been  reached  directly  from  the  sound  and  respectable 
desire  that  all  commodities  should  fall  in  cost- value  and  in 
esteem-value,  since  variations  in  the  latter  are  frecpiently,  and 
in  isolated  cases  almost  always,  accompanied  by  variations 
in  the  former.  Here  is  something  possible.  But  it  involves 
something  else,  which  also  is  often  ignored.  This  is  that  if 
every  commodity  becomes  cheaper  in  price,  that  is,  in  exchange- 
value  in  money,  money  becomes  dearer  in  exchange-value  ;  and 
if  the  former  is  still  desired  when  this  is  perceived,  the  latter 
must  also  be.  Why  are  these  two  things  to  be  desired  ?  The 
only  possible  reply  as  regards  the  first  is  that  the  desire  for 
the  fall  of  prices  is  conjoined  with  the  desire  for  the  fall  of  all 
■commodities  in  cost-value  or  in  esteem-value.  It  is  thought, 
with  or  without  good  reason,  that  the  desired  fall  in  these 
values,  if  occurring,  should  be  marked  and  measured  by  a  cor- 
responding fall  in  their  prices.  And  this  thought  necessarily 
involves  the  idea  that  money  is  to  be  considered  the  standard 
measure,  not  of  exchange-value,  but  either  of  cost-value  or  of 
esteem-value. 

It  is  not  within  the  province  of  this  work  to  argue  as  to 
whether  money  should  bo  considered  the  standard  of  the  one 
or  of  the  other  of  the  various  kinds  of  value.  But  it  is  essen- 
tial here  to  point  out  that  money  cannot  possibly  be  the  standard 
of  more  than  one  kind  of  value.  With  but  the  rarest  excep- 
tions, all  the  world  has  hitherto  been  agreed  that  money  is  a 
measure  of  "  value,"  and  consequently  that  in  order  to  serve 
this  function  properly  and  to  be  worthy  of  being  taken  as  a 
standard,  it  ought  to  be  stable  in  "  value."  It  becomes  ueces- 
.sary  now  to  distinguish  in  which  sense  the  terra  "  value  "  must 
be  understood  in  this  connection.  To  continue  speaking  of 
money  merely  as  the  measure  of  "  value  "  will  be  to  perpetuate 
an    equivocation    of  thought    tiiat   may    lend  aid   to   either  of 


PRl^T'II'r.ES    OF    (iltltKXCY    RE(JrLATI()N  489 

opposite  sides  on  many  important  practical  questions  according 
to  the  desires  selfishly  uppermost  in  the  minds  of  partisans. 
Here  is  a  point  which  all  economists  should  pause  to  decide 
and  settle,  before  they  attempt  to  take  a  further  step  in  their 
science — or  at  least  in  that  branch  of  it  which  is  the  science  of 
money. 

^  4.  As,  however,  this  work  specially  deals  with  exchange- 
value,  we  may  here  assume,  for  the  sake  of  theory,  that  money 
is  properly  the  standard  measure  of  exchange-value.  Then 
money  ought  to  be  stable  in  exchange-value  ;  and  if  it  is  not 
so  naturally,  it  ought  to  be  made  so  artificially,  if  this  be  pos- 
sible. And,  as  another  statement  of  the  same  thing,  the  gen- 
eral level  of  prices  ought  to  be  constant  ;  and  if  it  is  not  so 
naturally,  it  ought  to  be  made  so  artificially,  if  this  be  possible. 
And  this  in  spite  of,  and  through,  the  fact  that  it  is  desirable 
for  all  commodities  to  fall  in  cost-value  and  in  esteem-value. 

The  collective  will  of  the  people,  as  organized  in  government, 
cannot  properly  control  the  causes  at  work  upon  commodities, 
except  for  the  purpose  of  unfettering  and  aiding  production, 
and  of  making  all  commodities  fall  in  cost-value  and  in  esteem- 
value.^  In  fact,  the  fall  in  cost-value  is  the  aim  of  every 
producer  of  every  class  of  commodities,  while  the  fall  in  esteem- 
value,  aimed  at  by  the  consumers,  is  the  result  of  the  free  play 
of  all  these  agencies.^  But  it  is  believed  to  be  within  the  power 
of  government,  by  assuming  the  issuance  of  money,  to  control 
the  exchange-value  of  money  in  all  things, — that  is,  also,  the 
exchange-value  of  all  things  together  in  money, — without  in 
any  wise  seeking  to  control  their  relative  exchange-values 
amongst  themselves.  If  this  be  so,  the  aim  should  be,  neither 
to  make  money  cheaper  in  commodities,  that  is,  to  make  prices 
rise,  nor  to  make  commodities  cheaper  in  money,  that  is,  to 
make  prices  fall,  but  to  keep  money  stable  in  exchange-value  in 

*  Proper/ y,^aga,iu  explanation  is  neetled, — because  as  a  matter  of  fact  govern- 
ments do  exactly  the  opposite  and  raise  the  cost-value  and  esteem-value  of  goods 
— hy  taritfs,  keeping  out  the  goods  produced  more  cheaply  abroad  that  could  be 
exchanged  for  the  goods  produced  more  cheaply  at  home. 

■'  Without  this  free  play  the  producers  may  succeed,  through  monopoly,  not 
only  in  lowering  the  cost-value,  but  even  in  raising  the  esteem-value  of  their 
goods,  by  curtailing  production. 


4})()  UTIT.ITY    OF    .MONETARY    MEASUREMENTS 

coiuniotlities  and  coiiiiiiodities  as  a  whole  stable  in  money,  that 
is,  to  make  the  general  level  of  prices  constant,  so  that,  while 
the  esteem-values  of  all  commodities  are  happily  falling  with 
the  fall  in  their  cost-values,  the  esteem-value  of  money  shall 
fall  neither  more  rapidly  nor  more  sluggishly  than  the  esteem- 
values  of  all  commodities  on  the  average. 

Take,  for  instance,  the  case  above  supposed  of  one  commod- 
ity rising  50  per  cent,  in  exchange  value  in  another,  the  latter 
then  falling  33|^  per  cent,  in  exchange-value  in  the  former. 
Let  us  revive  our  earlier  supposition  of  a  simple  economic 
world  with  money  and  two  classes  of  commodities.  Now  if  the 
rise  of  [A]  by  50  per  cent,  in  [B]  were  manifested  in  prices 
by  [A]  rising  50  per  cent,  in  price  while  [B]  remained  con- 
stant in  price,  this  would  mean  that  the  average  of  prices  has 
risen  and  the  exchange- value  of  money  has  fallen  ;  which  is  a 
sign  that  money  is  too  abundant,  whatever  were  the  other  causes 
at  work.  Some  money,  therefore,  should  be  abstracted  from 
the  circulation.  This  would  have  the  effect  of  lowering  the 
prices  of  both  the  commodities  in  (about)  the  same  ratio.  If 
the  contraction  should  proceed  until  the  price  of  [A]  stood  at 
1.2247  above  what  it  was  at  the  start  and  the  price  of  [B]  at 
.8165  below  what  it  was  at  the  start,  we  know  by  our  investi- 
gations that  then,  on  the  supposition  of  the  two  classes  being 
equally  important  over  both  the  periods  together,  the  exchange- 
value  of  money  is  the  same  as  it  was  in  the  beginning.  On  the 
other  hand  if  the  relative  change  -between  [A]  and  [B]  had 
manifested  itself  by  [B]  falling  in  price  by  33^  per  cent,  while 
[A]  remained  constant  in  price,  this  would  constitute  a  fall  of 
the  average  of  prices,  indicating  a  rise  in  the  exchange-value  of 
money,  signifying  an  insufficiency  of  money,  whatever  be  the 
other  causes  of  these  changes,  and  therefore  giving  warning  of 
the  need  of  issuing  more  money  for  the  purpose  of  running  the 
price  of  [A]  uj)  to  1.2247  and  the  price  of  [B]  up  to  .8105 
compared  with  what  they  were  at  the  start.  Or  the  prices  may 
originally  have  changed  in  any  of  three  other  typical  ways. 
But  whatever  their  original  changes,  due  to  natural  causes,  as 
soon  as  it  is  discovered  that  they  are  such  as  to  constitute  a  fall 


PRiNcii'LKs  OF  ('^Kl;I•:^(•^■    liWiiLA  rioN  491 

or  a  rise  in  the  exchan<>;e- value  of  money,  the  issnance  should 
be  altered  so  as  to  make  both  the  priees  move  into  their  proper 
counterbalancing  positions. 

If  one  of  these  classes  were  more  important  than  the  other, 
their  counterbalancing  positions  would  be  different ;  but  these, 
too,  can  be  calculated,  and,  when  calculated,  their  attainment 
should  be  aimed  at  in  the  regulation  of  the  currency.  Unfor- 
tunately, however,  in  this  case,  unless  the  mass-(|uantities  are 
constant,  the  calculation  cannot  be  made  with  absolute  preci- 
sion. Also  this  is  so  when  we  are  dealing,  as  we  must  in  our 
complex  world,  with  many  and  variously  large  classes.  Yet 
we  know  that  still,  the  more  complex  the  world,  the  more  is  the 
use  of  the  properly  weighted  geometric  average  of  the  price 
variations  likely  to  approximate  to  the  truth.  Therefore  it  is 
still  possible  to  used  the  geometric  method  with  practically 
sufficient  accuracy. 

Thus  in  general,  in  any  complex  economic  world,  if  one 
class  of  commodities  rises  ^>»  per  cent,  (in  hundredths)  in  ex- 
change-value in  all  other  commodities,  then,  for  money  to 
remain  stable  in  exchange- value,  money  should  rise  in  ex- 
change-value in  the  other  commodities  from  unity  apprc»xi- 
mately  to  the  geometric  average  between  1  repeated  /•  times  and 
1  -\-  p,  which  is  '  '\/'^  1  -|-  p,  or,  to  use  another  form  of  expression, 

(1  -\-  p)  '  +  1,  (r  representing  the  number  of  times  all  the  other 
commodities  are  more  im])ortant  over  both  the  periods  com- 
pared than   the  one  class   in  question)  ;  wherefore  the   prices 

of  these  articles  shcuild  fall   approximatelv  by  1  —  ,,    ,         JL 

per  cent.,  and,  as  the  price  of  the  one  article  must  rise  to 
1  -\-  p  times  the  other  prices,  it  should  rise  approximately  to 

,,         -    '    ,  or  (1  +  y:») '+1,  times  its  former  price,  approximatelv 

(1  -}-  p)r+\'       \    ^  I  I        y  I        '    rr 

by  (1  -f  7>)'  +  '  —  1  per  cent.  And  reversely,  if  one  class  falls 
p'  per  cent,  in  exchange-value  in  all  other  commodities,  its 
price  should  fall,  not  by  p'  per  cent.,  but  approximately  by 

]  —  (1  —  p')  +»  per  cent.,   and   the   prices  of  the   others,   in- 


402  rnuTV  of  mo.nktarv  measirhmknts 

stead  of  remaining  luiclianged,  should    rise  approximately  by 

;,  ,^    '    —  1  i)er  cent.      Or,  further,  if  out  of  all  classes  of 

(1  -/)'  +  ' 

commodities  u"  in  number,  certain  classes  I  in  number  (in  both 

cases  including  the  repetitions  needed  to  represent  their  relative 

sizes  over  both  the  periods)  rise  in  exchange-value  in  all  other 

commodities  evenly  by  p  per  cent,  each,  or  unevenly  so  that 

the  properly  weighted  geometric  average  of  their  rises  is  j)  per 

cent.,  then  money  should  rise  in  exchange-value  in  the  others 

approximately   to  the  geometric  average  between   1   repeated 

I 
n"  —  I  times  and   ^  -\-  p  repeated  I  times,  that  is,  to  (1  +  p) "" 

times  _  its  former  exchange-value  in  them ;  wherefore  they 
shituld  fall  in  price  evenly  or  on  the  average  approximately  by 

1  —  r^ ^-  per  cent.,  and  the  /  articles  should  rise  in  price 

{\  +  p) n"  ^ 

evenly  or  on  the  average  approximately  by  (1  +  py~""  —  1  per 
cent.  And  reversely  if  the  /  articles  fall  p'  per  cent,  in  ex- 
change-value in  the   others,   money  should  fall  in   the   latter 

t 
approximately  to  (1  —  p')n"  of   its  former   exchange-value  in 

them  ;  wherefore  their  prices  should  rise  approximately  by 
..  ,,  '    —  1  per  cent.,  and  the  prices  of  the  J  articles  should 

{\-p')."  A  A 

I 

fall  approximately  by  1  —(1  —  p'y~n"  per  cent.  Therefore  in 
any  of  these  cases,  whetiier  addition  or  subtraction  of  money 
be  needed,  it  should  stop  when  these  prices  are  obtained.^" 

The  actual  operation  of  the  system,  however,  would  be  some- 
what different,  and  slmj)lcr,  though  less  precise.  We  never 
know  directly  how  much  one  or  more  articles  vary  in  exchange- 
value  in  another  or  others  ;  but  we  learn  this  only  from  their 

'"In  practice  things  would  not  work  so  smootlily  as  in  the  theory.  Prices 
would  not  be  raised  or  lowereil  all  in  exactly  the  same  proportion,  for  tiie  reasons 
stated  in  Note  1.  For  instance,  small  alterations  in  the  quantity  of  money  might 
perhaps  raise  or  lower  some  prices  which  were  just  on  the  verge  of  making  the 
jump  from  one  round  figure  to  the  next,  while  others  at  the  verge  of  falling 
might  merely  l)e  advanced  to  the  verge  of  rising,  or  reversely,  without  showing 
any  actual  variiition.  Still  the  principle  is  the  same,  and  the  addition  or  sub- 
traction of  money  should  stop  when  the  prices,  whatever  they  be,  approximately 
compensate. 


IMUNCII'LHS    OF    <ri;HKN(  V     ItKi  ilLA'lION  4J)o 

prices  already  iorined  under  causes  coming  from  money  as  well 
as  from  the  commodities  themselves.  And  we  do  not  generally 
know  the  causes  which  make  the  changes  either  between  the 
commodities  or  between  these  and  money  ;  nor  for  our  present 
purpose  is  it  necessary  that  we  should  know  them.  Their 
eflFects  are  manifested  in  the  subseiiuent  prices,  and  to  know 
these  is  sufficient.  Here  is  where  the  formuhe  for  finding  the 
new  level  of  all  prices  compared  with  the  former  level  come 
into  practical  use, — and  now  one  of  the  other  two  is  preferable 
to  the  geometric.  But  perfect  precision  in  regard  to  the  exact 
extent  of  a  variation  upward  or  downward  is  not  of  great  im- 
portance, provided  the  result  i)e  always  given  in  the  right  di- 
rection. If  the  level  of  prices  is  shown  to  have  risen,  it  is  a 
sign  that  the  quantity  of  money  is  too  great  and  should  be 
diminished  ;  and  if  the  level  of  prices  is  shown  to  have  fallen, 
it  is  a  sign  that  the  quantity  of  money  is  too  small  and  should 
be  increased.  Of  course  it  is  well  that  the  measurement  should 
show  whether  the  rise  or  fall  is  great  or  small ;  for  then  it  will 
be  known  whether  a  great  or  a  small  alteration  needs  to  be  made 
in  the  quantity  of  money.''  The  alteration  must  continue  until 
a  subsequent  application  of  the  formula  shows  that  the  original 
level  is  again  obtained.  The  possibility  of  any  great  variation 
in  the  level  of  prices  should  be  cut  off  by  making  the  measure- 
ments at  short  intervals.'^ 

^1  Walras  once  thought  that,  the  variation  of  the  level  of  prices  and  the  quan- 
tity of  money  in  a  country  being  known,  he  could  calculate  exactly  the  new 
quantity  of  money  needed,  his  conclusion  being  that  the  former  quantity  should 
be  multiplied  by  the  reciprocal  of  the  variation,  B.  <j9,  p.  17.  But  things  do  not 
work  with  such  exactness  because  (1)  the  principle  upon  which  his  calculation  is 
founded,  namely  that,  given  the  quantity  of  goods,  the  level  of  prices  is  exactly 
in  inverse  proportion  to  the  quantity  of  money,  is  not  true,  and  (2)  even  if  it 
were,  there  would  be  intervening  changes  going  on  in  the  goods  before  the  addi- 
tion or  subtraction  of  money  could  be  carried  out  or  before  its  influence  could  be 
felt.     Walras  has  since  modified  his  views  on  this  subject,  in  B.  70,  p.  476. 

^2  Walras  wants  the  determining  measurements  to  be  made  for  every  decade, 
i.  c,  for  an  industrial  cycle,  B.  (il,  pp.  <I,  is — this  is  too  long.  At  the  other  ex- 
treme Williams,  op.  cit.,  p.  289,  Fonda,  B.  127,  p.  Kw,  and  Whitelaw,  B.  130,  pp. 
23,  51,  want  the  measurement  to  be  made  daily — this  is  too  short.  They  were 
recommended  yearly  by  Lowe,  B.  8,  p.  279,  half-yearly  or  quarterly  by  Laves, 
op.  cit.,  p.  842,  and  Pomeroy,  B.  135,  p.  333,  and  monthly  liy  .levons,  oj}.  cit.,  p. 
330,  and  Parsons,  B.  136,  p.  137.  Perhaps  it  would  be  best  to  make  provisional 
measurements  montiily  or  quarterly,  and  definite  measurements  yearly. 


404  ITILITY    OF    MONETAUY    MEASll!  KM  KNTS 

>i  0.  On  the  other  hand,  let  ns  for  a  moment  suppose  tliat  as  a 
measure  of  "  value  "  money  is  to  be  considered  the  measure  of 
cost-value  or  of  esteem-value.  Then  either  the  })ayment  of 
contracts  may  be  regulated  by  the  ascertained  variations  of 
money  in  the  one  or  the  other  of  these  kinds  of  value,  or, 
the  preventive  system  being  adopted,  the  aim  to  be  held  in 
view  would  be,  by  a  similar  regulation  of  the  quantity  of  the 
currency,  to  keep  money  stable  in  one  of  these  values,  forcing 
the  prices  of  all  commodities  individually  to  rise  or  fall  accord- 
ing as  the  commodities  individually  rise  or  fall  in  cost-value 
(although  here  there  would  be  difficulty),  or  in  esteem-value. 
Cost-value  being  selected  as  the  norm,  the  cost  of  production  of 
the  money-metal  (for  here  the  cost-value  of  paper  money  is  out 
of  the  question)  ought  on  the  average  to  be  constant. ^^  In  the 
case  of  esteem-value  being  chosen,  the  gross-money  earnings  of 
an  hour's  labor  of  all  the  workers  in  a  country  (or  their  net 
money-earnings,  along  with  the  money-incomes  of  an  hour,  in  a 
day  of  the  number  of  hours  the  others  work  on  the  average,  of 
all  the  non-workers)  ought  to  be  constant.  The  former  of  these 
standards  has  never  been  suggested  either  as  guide  for  the  pay- 
ment of  money-contracts  or  as  guide  for  regulating  the  issuance 
of  money — the  last  being  possible  only  by  interference  with  the 
working  of  the  mines  of  the  precious  metal  or  metals,  which  has 
never  been  recommended  in  this  connection.  The  latter  stand- 
ard has  been  suggested  for  both  these  uses.  The  suggestion 
has  been  made  both  that  contracts  should  be  payable  in  the 
quantity  of  money  that  is  constant  in  esteem-value  (imperfectly 
measured  by  wages  only),^^  and  that  the  issuance  of  money 
should  be  regulated  so  as  to  keep  money  stable  in  such  value 
(likewise  imperfectly  measured  by  wages  only).'"'  In  either 
case,  as  remarked  in  an  earlier  chapter,  after  having  made  such 
measurement  in  regard  to  money  alone,  if  we  have  a  desire  to 
know  whether,  and  how  much,  commodities  in  general  have 
risen  or  fallen  in  cost-value  or  in  esteem-value,  we  could  calcu- 
li On  the  (tvenige,  auil  not  at  the  least  fertile  iiiiiu',  Iieciuise  we  are  now  not 
dealing  with  relations  of  exchange  or  of  cstoeiii. 
1''  By  Shadwell,  Principles,  p.  2i)(i. 
^■'  Botli  indifferently,  l>y  Pollard,  op.  cil.,  pp.  74-7'). 


PRINCIPLES    OF    (TRKKNCV     UKc;  ILA'IK  »N  4!l.") 

late  this  (in  cost-value  only  imperfectly)  by  inverting  tlie  cal- 
culation of  the  variation  of  such  money  in  exchange-value  in 
all  commodities.  Thus  in  these  cases  also  the  measurement  of 
the  general  exchange-value  of  money  would  still  be  useful. 
Only  now  its  use  would  be  merely  theoretical  and  to  satisfy  an 
idle  curiosity,  while  the  really  important  measurement  would 
be  the  measurement  either  of  cost-value  or  of  esteem-value. 
But,  on  the  contrary,  if  money  is  properly  the  measure  of  ex- 
change-value, it  is  these  other  measurements  that  are  practically 
useless  and  serve  only  to  please  our  vanity  by  showing  that  we 
are  making  progress. 

§  6.  Both  the  above  discussed  practical  schemes  for  making 
use  of  the  measurement  of  variations  in  the  general  exchange- 
value  of  money,  or  in  the  general  level  of  prices,  are  still  em- 
bryonic, and  no  attempt  to  apply  them  will  probably  be  made 
for  centuries  to  come."'  But  a  stumbling  block  in  their  way  is 
the  fact  that  the  measurement  of  exchange- value  has  never  been 
perfected  so  as  to  win  unanimous  assent  on  the  part  of  econ- 
omists, as  is  necessary  before  scientific  knowledge  can  be  claimed 
concerning  the  very  variations  which  those  schemes  propose  to 
correct  or  to  prevent.  Also  this  lack  of  science,  which  is  still 
greater  in  regard  tq  cost-value  and  to  esteem-value,  is  no  doubt 
a  reason  why  economists  have  not  turned  their  attention  to  a 
more  careful  consideration  of  the  (juestion  whether  money  is  the 
measure  of  exchange- value  or  of  cost-value  or  of  esteem-value. 

^''  Of  course  the  mensuration  of  exchange-value  should  not  be  postponed  until 
practical  use  can  be  made  of  it ;  for  practical  use  can  be  made  of  it  only  after  it 
has  been  performed.  Hence  it  is  advisable  even  now  that  in  the  mass  of  statistics 
which  every  civilized  government  makes  it  a  duty  to  collect  there  should  be  an 
effort  made  to  measure  the  variations  in  the  exchange-value  of  money.  The  cre- 
ation of  an  official  bureau  for  this  purpose  has  been  recommended  by  Scrope,  B. 
9,  p.  424;  Newcomb,  B.  76,  p.  213 ;  Marshall,  B.  93,  p.  365  ;  British  Association 
Committee,  Fourth  Report,  B.  102,  p.  487 ;  Laves,  op.  cit.,  p.  S46  ;  Zuckerkandl, 
B.  11.5,  p.  249. 


APPENDIX    A. 

ON    VARIATIONS   OF   AVERAdES   AND   AVKHACJES  OF 
VARIATIONS. 

The  supposition  is  tliat  we  have  two  sots  of  Hgures,  a^,  6„  ,  at  a  first 

moment  or  period,  and  at  a   hiter  moment  or  period  certain  other  figures, 

Oj,  62,  ,  to  wliich  those  have  varied.'     We  may  have  any  numbers  of 

these  figures  at  botli  periods,  but  always  the  same  number  in  each  set  ;  for 
otherwise  we  should  have  a  figure  in  the  first  which  does  not  vary  to  anything 
in  the  second,  or  a  figure  in  the  second  to  which  nothing  in  the  first  has  varied. 
The  number  of  the  a's,  whether  one  or  more,  may  be  represented  by  x,  that 
of  the  6's  by  ij,  and  so  on.     For  convenience,  although  this  phraseology  is  not 

strictly  accurate,  we  may  speak  of  the  symbols  a,,  02,  6,,  b.^,  ,  as  figures, 

and  of  the  symbols  .c,  y,  ,  as  numbers  (this  referring  to  the  numbers  of 

times  the  figures  occur  in  each  set).  The  figures  constitute,  so  to  speak, 
classes,  there  being  as  many  classes  as  there  are  distinct  figures.  We  may 
have  reason,  provided  by  the  problems  we  are  dealing  with,  to  divide  even 
similar  quantities  into  distinct  classes,  and  tiicn  to  represent  them  by  distinct 
figures.^ 

The  figures,  of  course,  are  quantities.  But  sometimes  it  will  be  convenient 
to  speak  of  their  sizes,  when  it  is  desired  to  call  attention  to  their  quantitative 
relations.  The  classes  also  have  sizes,  made  up  of  their  figures  multiplied  by 
their  numbers, — e.  g.,  xa^,  .ra.2,  yb^,  etc. 

The  variations  of  the  figures  from  the  first  to  the  second  period  are  repre- 
sented by  the  exi)ressions    '",  ,'■' , The  variation  of  «,  to  a.,  is  a  variation 

of  rt,  bv     "  ;  for  (I,  ■    '      «...      It  is  the  same  also  as  a  variation  of  1  I  ^    '  )  to 

^ .     Of  course  the  variation  of  the  whole  class  .cai  to  xa.^  is  nothing  else  as  a 

a, 

variation  ;  for'    ^         ■' .     But  the  variations  of  all  the  figures  in  a  class,  thus 
..0,       o, 

^If  <(2=«i,  this  may  he  viewed  as  an  infinitt-ly  small  variation.  Therefore 
we  may  include  it  in  speaking  of  variations. 

-If  fti  =  61,  but  a.,  and  /ja^re  unequal,  or  if  a.^  =  ^21  hut  ((1  and  /(jare  unequal, 
this  is  already  sutticieut  niathenuitical  reason  for  putting  the  (juantities,  though 
the  same  at  one  period,  into  different  classes.  But  even  if  (ii  -  /»,  and  rtj  =  b.^, 
tliere  may  he  other  reasons  for  treating  them  as  belonging  to  distinct  classes. 
Though  classes  properly  contain  many  individuals,  single  figures  without  mates 
may  still  he  viewed  as  constituting  classes. 
32  497 


498  APPENDIX    A 

'■',     - ,  to  .1-  terms,  when  added  together,  are  .r    ^,  or  when  multiplied  hy 

Qj     a,  O) 

one  another,  arc  I     '  I  . 
\"i  / 

We  may  average  each  set  of  hgures  seiiarately.     Representing  the  average 

of  the  figures  at  the  first  ))erio<I  hy  A,,  and  that  of  the  figures  at  the  second 

A, 

A,' 

variation  departing  from  constancy  as  this  expression  departs  from  unity.     Or 

we  may  average  the  variations  of  the  figures  as  they  change  from  what  they 

were  at  the  first  period  to  what  they  become  at  the  second.     Representing  this 

A, 
by  V,  we  shall  find  sometimes  that   .  '   -  V,  and  sometimes  that  these  expres- 

Ai 

sions  are  difi'erent.      It  is  our  purpose  to  study  the  relationshijj  between  these 

forms. 

In  averaging,  to  assign  a  certain  importance  to  one  term — -a  class  of  figures, 
or  a  kind  of  variations — relatively  to  others,  is  to  weight  it.  The  most  natural 
way  to  weight  classes  of  figures  is  by  the  numbers  of  times  the  figures  occur 
in  each  class.  Similarly  it  would  seem  most  natural  to  weight  kinds  of  vari- 
ations by  the  numbers  of  variations  in  each  kind.  But  as  the  figures  may  be 
reported  capriciously,  and  the  steady  element  be  the  sizes  of  the  classes,  w^e 
shall  find  the  need  of  weighting  the  variations  according  to  the  latter,  in 
several  ways. 

The  number  of  all  the  classes,  and  conse<jueutly  of  all  tlie  kinds  of  vari- 
ations, will  be  represented  by  //,  sotliatw  1  for  k^,  a.,,  or  "'  -f-  1  I  for  6,, 
h.,,  or  "  -f-  tlirough  the  whole  lists.  The  number  of  individual  fig- 
ures at  each  period,  and  the  number  of  tiie  individual  variations,  will  be  rep- 
resented l>y  ??',  so  that  71'' =  .»:-(- 1/ -|-    tonterms.     If  there  is  only  a  single 

figure  in  every  class,  then  n^  -  -n  ;  otherwise  n^  >  n.  Thus  »/  represents  the 
sum  of  what  we  have  just  described  as  the  most  natural  weights  of  the  classes 
of  figures  we  are  dealing  with.  The  sum  of  the  other  weights  which  we 
shall  need  for  the  variations  (and  sometimes  for  the  figures)  will  always  be 
represented  by  //',  although  these  weights  are  different  on  difierent  occasions. 

Before  proceeding  to  our  sui)ject  proper,  it  may  be  well  to  state  certain 
[)rinciples  which  are  coitniion  to  all  kinds  of  averages. 

1.    ON   AYKUAGKS   IN   GI:NEIIAL. 

'i  1.  All  average  of  any  set  of  terms  is  always  smaller  than  the  largest  and  larger 
than  the  smallest,  whatever  be  the  number  of  terms.     Consequently 

'i  2.  //'  llie  terms  are  all  equal,  the  average  is  equal  to  each  term,  whatever  be 
their  number. 

These  two  principles  are  explicative,  that  is,  they  flow  from  the  defini- 
tion of  "average,"  and  no  formula  pretending  to  be  a  formula  of  a  kind  of 
average  is  so  unless  it  yields  results  in  accordance  with  these  principles.  That 
the  formula'  for  the  arithmetic,  harmonic  and  geometric  averages  satisfy  the 


VAiMATioNs   AND   A \' i:i;a(; KS  499 

second  priiicipK'  li;is  incidt'iUnlly  heon  sliown  in  ('liai)ti'r  V.  Sect.  I.  ^^5 
and  6. 

Consequently  also,  if  we  are  "dealing  with  sets  of  terms  in  classes, 

^  3.  Tlie  avercuje  is  always  between  the  highest  and  the  lowest  terms  appeAirincf  in 
any  of  the  classes,  whatever  be  the  numbers  of  terms  in  any  classes  ;  and 

^4.  If  the  terms  ore  alt  equal,  but  arranged  in  classes  with  different  numbers  in 
each,  the  average  is  equal  to  each  of  the  figures,  whatever  be  the  numbers  of  the 
figures  in  any  of  the  classes. 

A  meaning  in  this  last  proposition  is  that  the  statement  is  luiiversally  true, 
whatever  be  the  weighting  of  tlie  classes.  In  other  words,  when  all  the  terms 
are  equal,  the  weighting  is  indifierent. 

'i  5.  What  has  baen  said  of  averaging  any  terms  is  true  whether  the  terms 
be  figures  (as  above  described)  or  variations  (of  iigures).  For  example,  ap- 
plied to  variations,  the  fourth  pi'oposition  becomes  :  If  the  variations  are  all 
equal,  the  average  is  equal  to  each  of  them,  no  matter  what  be  the  weighting  of  the 
variations. 

That  the  formube  for  the  three  averages  satisfy  this  principle  will  be  indi- 
cated in  the  course  of  the  separate  treatment  of  them. 

^  6.  Furthermore,  if  the  variations  are  all  equal,  the  variation  of  the  aver- 
ages separately  drawn  of  the  figures  which  have  varied  and  of  the  figures  to  which  thty 

have  varied  (    .^  )  is  the  same  as  the  average  of  the  variations  (V),  itself  the  same 

,   ,'       .     .        /  «.  \ 

as  any  one  oj  the  variations  ( ''.  g.     "  1 . 

If  all  the  figures  in  each  set  are  equal,  this  is  self-evident.  In  other  cases 
a  demonstration  may  be  needed  for  each  kind  of  average.  Indications  of  the 
demonstration  will  be  added  in  the  treatment  of  each  kind  of  average.  (See 
below,  notes  in  II.  <;  6,  III.  ^  6,  and  V.  ?  5). 

Of  course,  if  the  variations  are  all  ^,  or  1,  that  is,  if  there  are  no  varia- 
tions, this  principle  applies,  and  is  now  self-evident,  whatever  the  figures. 
For  in  this  case  the  corresponding  figures  in  the  two  sets  are  identical,  the 
sets  themselves  are  so,  and  so  must  be  their  averages  ;  wherefore  there  is  no 
variation  of  the  averages. 

We  shall  be  interested,  in  what  follows,  principally  with  sets  of  figures  in 
at  least  one  of  which  at  least  one  of  the  figures,  and  one  of  the  variations,  is 
different  from  the  rest — or  rather,  with  general  conditions  which  admit  the 
possibility  of  such  divergence. 

§  7.  Even  now  there  is  another  case,  common  to  all  the  kinds  of  averages, 
in  which  the  variation  of  the  averages  and  the  average  of  the  variations  always 
agree.  If  all  tiic  figures  at  the  first  period  are  units,  wherefore  the  average  also 
is  a  unit,  the  figures  in  the  second  set  themselves  express  their  variations  from 

the  first  period,  since  in  all  the  expressuins  tor  tiie  variations,       ,       ,  , 

the  denominatoi-s  are  1,  and  may  be  dropped.     Therefore  the  expression  for 

A, 

the  variation  of  the  averages,       " ,  in  which  the  denominator  is  also  1,  and 

may  be  dropped,  and  in  whicli  the  numerator  is  the  average  of  tlie  figures  at 
the  second  period,  is  also  the  expression  for  the  average  of  the  variations. 
Thus 


oOO  AIM'HNDIX    A 

If  (til  thefigure.<  at  the  first  perml  are  units,  the  variation  of  lh<  avenu/es  and  the 
uvetxKje  of  the  variations  are  alwai/s  the  saiw. 

This  proposition,  liowever,  is  very  different  from  tiu-  preceding.  In  the 
conditions  there  jjosited  it  is  indifferent  what  weighting  is  used.  Here  it  is 
importimt  that  correct  weighting  he  used,  and  it  is  necessary  that  the  same 
weighting  be  used  in  avei-aging  the  figures  (of  the  second  period)  and  in  aver- 
aging the  variations. 

Moreover  this  proposition  is  true  also  of  all  cases  when  the  fi(jiurs  at  the  first 
period  arc  equal,  whether  they  all  be  units  or  any  other  (piantity.  For  this 
other  quantity  may  be  taken  as  a  unit,  and  the  figures  in  the  second  set  be  re- 
duced on  the  same  scale,  and  then  the  proposition  applies. 

Consequently  this  principle  will  form  part  of  all  our  separate  treatment  of 
the  three  kinds  of  averages  (except  in  the  third,  Avhere  it  will  be  swallowed 
up  in  a  wider  principle).  The  particular  demonstrations  will  tlicrefore  be 
given  later.     (See  II.  §  6,  III.  §  6,  V.  §  5). 

Applying  to  all  cases,  the  following  principles  are  also  plain. 

§  8.  Given  a  set  of  figures  (or  variations)  not  all  alike,  of  which  the  aver- 
age (of  any  kind)  is  known,  if  we  add  to  it  a  figure  equal  to  the  average,  or 
substract  from  it  a  figure  which  happens  to  be  equal  to  the  average,  we  do  not 
alter  the  average.  Hence  it  does  not  matter  how  often  a  figure  equal  to  the 
average  be  added  or  subtracted,  or  whether  it  be  omitted  altogether.  In  other 
words,  the  loeix/hting  of  a  figure  equal  to  the  average  is  indifferent. 

The  similarity  of  this  to  §  4  is  patent. 

>/.  i».  Given  a  set  of  figures  (or  variations)  as  before,  if  we  add  a  figure  larger 
than  the  average,  we  raise  the  average  somewhat ;  and  if  we  subtract  such  a  fig- 
ure we  take  away  one  of  the  inHuences  that  have  made  the  average  as  high  as 
it  is,  and  so  we  lower  the  average.  Hence  it  does  matter  how  often  such  a 
figure  is  repeated,  and  to  increase  the  number  of  times  it  is  repeated  or  to  en- 
large its  weight,  is  to  raise  the  average  nearer  to  it,  and  nearer  and  nearer  the  more 
we  repeat  it  (but  the  average  can  never  reach  it,  short  of  an  infinite  number 
of  repetitions,  for  in  that  case  its  weight  would  be  indifferent);  and  to  de- 
crease the  number  of  times  it  is  repeated,  or  to  diminish  its  weight,  is  to  lower 
the  average  (until  the  figure  is  omitted  altogether,  when  the  average  is  what  it 
would  liave  been  without  this  figure). 

^10.  Given  a  set  of  figures  (or  variations)  as  before,  if  we  add  a  figure 
smaller  than  the  average,  we  lower  the  average  somewhat ;  and  if  we  subtract 
such  a  figure  we  take  away  one  of  the  infiuences  that  have  made  the  average 
as  low  as  it  is,  and  so  we  raise  the  average.  Hence  it  does  matter  here,  too, 
how  often  such  a  figure  is  repeated,  and  to  increase  the  number  of  times  it  is 
repeated,  or  to  enlarge  its  weight,  is  to  lower  the  average  nearer  to  it,  and  nearer 
and  nearer  the  more  we  repeat  it  (btit  without  ever  reaching  it,  as  before); 
and  to  decrease  the  number  of  times  it  is  repeated,  or  to  diminish  Us  weight,  is 
to  raise  the  average  (until  the  figure  is  omitted,  when  the  average  is  what  it 
would  have  been  without  this  figure).' 

'  Of  course,  if  we  add  or  subtract  a  figure  equal  to  one  average  and  unetiual  to 
another  average,  wc  alter  that  other  average.  Or  if  we  add  or  subtract  a  figure 
unequal  to  this  average,  and  so  alter  this  average,  we  may  perhaps  not  atlect 
another  average  (to  whieli  this  figure  may  happen  to  be  equal).  We  are  dealing 
witli  eaeli  kind  of  average  cairiiil  tlirougliout. 


VAlMA'lloNS    .\M>    a\i-;i!A(;ks  ">()1 

These  three  priiu'.iples  can  i-asily  be  (U'nionstratetl  in  tlie  ciise  of  eacli  of  tlie 
tliree  kinds  of  averages.     But  they  are  too  phiin  to  need  demonstration. 

A  resume  of  them  is  that  wlien  a  figure  is  the  same  as  the  average  its 
weighting  is  indiflerent ;  otherwise  its  weighting  counts,  and  an  alteration  in 
the  weighting  of  the  classes  without  any  variation  in  the  sizes  of  the  figures 
(but  like  a  variation  in  the  size  of  the  figures)  causes  the  average  to  change, 
the  influence  of  an  alteration  in  tlie  weighting  being  different  according  as  the 
figure  operated  on  is  larger  or  smaller  than  the  average.  To  increase  the 
rejietitions  of  a  larger  figure  has  the  same  influence  as  to  enlarge  the  figure  ; 
and  to  increase  the  repetitions  of  a  smaller  figure  lias  the  same  influence  as 
to  diminish  the  figure  ;  and  conversely. 

A  practical  application  of  these  three  principles  is  that  an  error  in  our 
weighting  of  a  figure  equal  to  the  average  is  of  no  account  (r.  e.,  if  the  aver- 
age without  a  figure  is  the  same  as  the  avei'age  with  it,  we  need  not  concern 
ourselves  about  its  weight).  And  the  nearer  a  figure  is  to  the  average,  the 
less  an  error  in  its  weighting  will  count ;  and  the  more  it  will  count,  the  more 
the  figure  is  removed  from  the  average.  Of  course  in  the  case  of  an  average 
of  variations,  what  is  here  said  applies  to  a  variation  according  as  it  is  the 
same  as,  near  to,  or  far  from,  the  average  of  the  variations. 

Hence  lastly, 

§  11.  If  all  the  weiyht.-i  arc  (ilkrcd  in  the  same  proportion,  there  w  no  effect  upon 
the  average,  or,  in  other  words,  if  we  have  the  proper  weights,  we  may  alter 
them  as  we  please,  so  long  as  we  keep  them  in  the  same  proportion  (multi- 
plying or  dividing  them  all  by  the  same  quantity). 

For  if  we  increase  all  the  weights  in  the  same  proportion,  the  increase  of 
tiie  weights  of  the  figures  equal  to  the  average  has  no  influence,  and  the  in- 
crease of  the  weights  of  the  figures  larger  than  the  average  tends  to  raise  the 
average,  while  the  increase  of  the  weights  of  the  figures  smaller  than  the 
average  tends  to  lower  it ;  but  as  the  influence  of  all  the  figures  below  the 
average  to  lower  it  is  equal  to  the  influence  of  all  the  figures  above  the  aver- 
age to  raise  it  (for  otherwise  the  average  would  not  be  where  it  is),  so  the  in- 
fluence of  the  changes  in  the  numbers  of  the  former  is  equal  to  the  opposite 
influence  of  the  proportionally  equal  changes  in  the  numbers  of  the  latter, 
and  the  average  remains  where  it  was.  And  reversely  if  we  decrease  all  the 
weights  in  the  same  proportion. 

If  this  general  explanation  be  not  sufficient,  the  proposition  may  be  dem- 
onstrated in  the  case  of  each  of  the  averages  separately.  This  will  be  done 
incidentally  for  two  of  them  in  the  following  pages.     (See  I.  §  7,  II.  §  7). 

Or  another  general  proof  may  be  made  as  follows.  If,  for  instance,  we 
double  all  the  weights,  we  may  segregate  all  the  new  terms,  and  so  form  two 
distinct  sets  of  figures  (or  variations)  exactly  duplicating  each  other,  one  of 
them  being  the  original  set.  Then  the  average  of  each  set,  separately  drawn, 
will  be  the  same,  ('onseijuently  the  one  average  of  the  two  together  will  be 
the  same.     And  similarly,  wiiatever  be  the  multiplier. 

Thus  the  general  system  of  weighing  depends  upon  the  relative  sizes  of 
the  weights,  and  not  upon  tlieir  absolute  sizes.  However  large  or  small  the 
weights,  if  in  the  same  proportion,  we  practically  have  the  same  weighting. 

This  being  so,  we  have  even  weightimj  whether  we  count  every  figure  in  the 
sets  (or  every  variation)  only  once,  or  an  equal  number  of  times. 


502  APPENDIX    A 


ir.    AUfTHMETIC   AVERAGIXU. 
§  1.   Witli  siiisik'  ti.<,'urt's  in  the  two  sets,  the  arithmetic  average  at  the  first 
period  is  -  (a,  -|-  />,  + to  /*  terms),  and  at  the  second  it  is  -  (aj-f-ij  4- 


to  n  terms)  ;  wliercfore  tlie  variation  of  the  averages  is 
A.      i  (»>  +  '=  + ) 


•*•'       i  (",  +  »,  + )' 


which  reduces  to 


A,       rt„  +  6^+-  .^».^ 

A,  ^~  «i  +  'j,  4- ' 

(thus  showing,  incidentally,  that  it  is  indifferent  whether  we  compare  the  arith- 
metic averages  or  the  sums). 

Here  the  weighting  of  tlie  figures  in  each  average  is  even,  each  figure  count- 
ing once. 

§  2.  Given  the  same  single  figures,  the  arithmetic  average  of  the  varia- 
tions, likewise  with  even  weighting,  each  variation  counting  once,  is 


v^i  ("'  +  »'+  ). 


This  is  a  difl'erent  expression,  not  universally  reducing  to  the  preceding. 
§  3.    With  cbisses  of  figures,  the  arithmetic  average  at  the  first  period  may 
be  expressed  in  full  tlius, 

Aj  =    ,  I  (f'l  +  «i  ~l~ t<)  '.r  terms)  +  (^^  +  ''i  +  to  _y  terras)  + 

to  «  classes  I , 

which  may  be  abl)reviated  to    ^  (.nr,  -\-  tJb^  -{- to  /(  terms),  or  may  also  be 

xa, -{- ifb, -\- 1      ,     ,  1  1  •     1    .     1-1 

written '- — , .     And   tlie  average  at  the   second   period   is  like 

■^  +  y  + 

unto  it,  with  change  only  in   the  numbering  of  a  ,  b , Therefore  the 

variation  of  the  averages  is 

A        n^  ^''"^  +  y^2  +  to  II  terms) 

'  /  ( -I'^i  +  y'*!  4- to  w  terms ) 

which  i-fduces  to 

Aa ■'■«2  +  .'/''•.'  4-    

Ai"~.mi  +!lbi  4- ' 

(in  which,  again,  the  variation  of  tlic  arithmetic  averages  is  the  same  as  tlie 
variation  of  the  sums). 

Here  the  weighting  of  the  figures  in  each  average  is  directly  according  to 
the  numbers  of  times  the  figures  are  repeated  in  each  class.  Naturally  an 
average  of  the  figures  taken  only  once  each,  as  in  §  1,  is  not  an  average  of  the 
figures  here  supposed,  but  only  of  the  figures  there  supposed.  ( Yet  the  evenly 
weighted  average  might  l)e  said  to  be  the  average  of  the  classes  simply  as 
classes,  each  class  counting  as  an  individual,  w  itliout  regard  to  the  numbers 
of  figures  in  them) 


VARIATKtNS    AND    AVKRACiKS  50.'J 

§4.   Given  the  same  classes  of   titjures,  tlie   similarly  weiirlited  aritlinictic 
average  of  the  variations  in  full  is 

V  ^  ;7  {(I!;  + 1;  + '"  •'•  ^-•'"^)  +(';  +  !:;* •" "  •^■^'"^) 


whieli  inav  be  abbreviated  to  — ;  (  .r    "  -|-  */ ,'"  -f- 


•••   to  /(  classes 
to  a  terms  |  ,  or  mav  he 


written 


y  «'.  ''. 

■'•  +  II  + 


Again  this  expression  is  different,  not  universally  reducing  to  tiie  preceding. 

§  5.  When  dealing  with  single  figures,  we  may  say  that  we  are  dealing 
with  classes  in  each  of  which  there  is  only  one  individual.  Therefore  we  may 
subsume  sets  of  single  figures  (or  variations)  under  sets  of  classes  of  figures 
(or  variations),  and  treat  only  of  the  latter.  Thus  in  both  the  preceding  cases 
we  have  been  dealing  with  arithmetic  averages  of  classes  in  which  the  weight- 
ing is  according  to  the  numbers  of  figures  in  the  classes,  and  of  averages  of 
variations  with  weighting  likewise  according  to  the  numbers  of  varying  figures 
in  the  classes. 

The  conclusion  from  the  preceding  paragraphs  then  is  that  the  variation  of 
the  arithmetic  averages  and  the  arithmetic,  average  of  the  variations,  in  all  cases  with 
weighting  according  to  the  numbers  of  figures  in  the  classes,  are  not  universaUi/  Ihe 
same. 

^  6.  The  variation  of  the  arithmetic  averages  and  the  arithmetic  average  of  the 
variations,  all  with  weighting  according  to  the  numbers  of  figures  in  the  classes,  are 
the  same  when  all  the  Jigures  at  the  first  period  are  equnl. 

Let  r(j=:6j= — -s  (s  being  am/  figure  above,  below,  or  at  unity,  in- 
tegral or  fractional).    Then  the  expression  for  the  comparison  of  the  averages 

(that  for  the  sums)  in  ^3  reduces  to   —-  t^~- t  ;  and  the  expression 

•  s('-|-.'/+ ) 

for  the  average  of  the  variations  in  §  4  reduces  to  the  same.     Q.  E.  D. 

Hence  if  we  reduce  all  the  actual  figures  in  an  irregular  set  at  the  first 
period  to  the  same  ideal  figure  (by  taking  some  common  divisor  of  them) 
reducing  the  figures  at  the  second  period  in  the  same  proportions,  and  if  we 
change  the  numbers  of  the  figures  in  the  classes  in  the  inverse  proportions  (so 
as  to  keep  unchanged  the  sizes  of  the  classes),  the  weighting  desired  in  order 
to  make  the  two  methods  agree  will  be  in  accordance  with  the  adjusted  num- 
bers of  these  ideal  individuals  in  the  classes. 

Thus  a  factor  which  affects  the  variation  of  the  averages  is  the  number  of 
times  the  figures  occur  in  each  class,  or  weighting  of  the  variations  directly 
according  to  the  niniibers  of  the  figures. 

In  all  other  cases  (except  of  course  when  all  ^the  variations  are  equal — 
see  I.  §  6 '  )  the  average  of  the  variations  will  require  diflerent  weighting  of 
the  variations  to  make  it  agree  with  the  variation  of  the  averages. 

1  In  the  foriinila  above  given  in  i;4let     "  =  ,^  = =  r,  and  the  expression 

reduces  to  /•.     Now  a^  ^  ''rtj,  .V2  =  '''ii  ^Q^  ''O  on.     Introduce  these  into  the  for- 
mnla  in  ?  3,  and  tliis  also  reduces  to  r. 


504  APPEXJ)IX    A 

j)  7.  Whi'n  the  numhers  of  the  figures  in  the  c/a.s,ses  are  all  equal,  the  rariation  of 
the  arithmetic  averages  {each  with  weight imj  according  to  the  numbers  of  figures  in  the 
classes — in  this  case  with  even  weighting)  is  the  same  as  the  arithmetic  average  of  the 
variations  with  weighting  according  to  the  sizes  of  the  figures  at  the  first  period. 

Let  X  -—y^ =<•  Tlien  the  expression  for  tlie  variation  of  the  aver- 
ages reduces  to  4"'- — J^v  i  >  ^"^^  this,  bv  dropping  tlie  /  from  both  sides 

of  the  fraction,  to  the  expression  for  the  variation  of  the  figures  taken  singly, 
i.  e.,  with  even  weighting.  We  want  now  to  prove  that  this  expression  is  the 
same  as  tlie  expression  for  tlie  average  of  the  variations  with  weighting  ac- 
cording to  «,,  ij, 

The  expression  for  the  average  of  tlie  variations  witli  weighting  according 

to  a,,  />,,   is    -7.  (  a,  ""  +  /*,  ''^  + to  /(  terms  )  in  wliich  n"  r-a^  -\- 

ii  -(- to  ))  terms.     Tliis  expression,  l)y  reducing,  and  restoring  tlie  vahie 

of    n",   becomes  -^ ^ ,   which  is  the  same  as  the  expression  for  the 

"i  +  ii-l 

variation  of  the  averages.     Q.  E.  D. 

Hence  a  factor  whicli  affects  the  variation  of  tlie  averages  is  the  size  of  the 
figures  in  the  classes,  or  weighting  of  the  variations  directly  according  to  the 
sizes  of  the  figures. 

The  reason  for  this  it  is  important  that  we  should  perceive.  Take  the 
simple  case  of  two  figures  in  each  set,  or  two  reported  variations.  Suppose  a^ 
^s  larger  than  6,  by  an  integral  number  of  times,  say  q.  Xow  the  single  vari- 
ation of  rt]  to  «.,,  compared  with  the  single  variation  of  6,  to  h.,,  is  a  variation 
not  merely  of  a  q  times  greater  quantity,  but  of  q  times  more  quantities. 
Hence  it  virtually  contains  q  times  more  variations.  Therefore,  as,  according 
to  the  preceding  proposition,  the  average  of  classes  of  variations,  when  the 
figures  at  the  first  period  are  equal,  must  be  drawn  with  the  classes  weighted 
according  to  the  numbers  of  figures  in  them,  if  in  the  present  case  the  weight 
1    be  given  to  the  figure  6,,  the  weight  q  sluvuld   be  given  to  the  figure  rtj,  as 

really  being  a  class  composed  of  q  /)j's,  each  of  which  varies  by    "'.     Tn  fact, 

"i 
the  formula  for  the  variation  of  the  averages  may  be  analysed  into  this, 

/  ,    a.,   ,    ,    a.,    ,  \    ,    ,    /'•> 

6,    "-ff),    "  + to  7  terms     + /),    - 

( />,  -|-  />,  + to  q  terms )  +  ''i 

wliicli  reduces  to  — ,^S^r^'       ;»ud  to     ,,  (  o    '"' -|-  ,'  ),  wliicli  is  the  exi)res- 
6,(9 -fl)  //'  V    "i        /'i  / 

sion  for  the  average  of  the  variations  witb  q  and  1  for  tiie  weights.  What 
is  here  shown  of  two  figures  so  conveniently  related  may  l)e  generalized  as  fol- 
lows. Having  one  reported  variation  of  n,  to  a-i,  one  reported  variation  of  6j 
to  6^,  and   so   on,  we  may  view  the  variation   of  ",    to  a.^  as  consisting  of  a-^ 

variations  of  1   (  '  |  to    '"' ,  and  the  variation  of  U.  to  b.,  as  consisting  of  /), 

V       ",  /        "i 

Viiriiitiuiis  of  1  (  '   I  to  ,"',  and  so  (111  with  all  the  other  reported  single  varia- 

V        b,  }        b. 


VAKIATIONS    AM)    A  V  Ki;  A( ;  KS  ,")(),") 


tions.      Tlu-n  tlie  weip;lit  of",  v:iri;itii)iis  df  1  to    "  is  n,,  and  tliat  of /<,    varia- 

"i 

tions  of  1  to     "'  is  i,,  and  similarly  in  all  the  other  cases;  and   tlu-   rigorous 

expression  for  the  arithmetic  average  of  these   variations,  according  to  §3, 

is  as  above  given. 

This  way  of  viewing  tiie  variations  is  a  perfectly  correct  way  ol  viewing 

them,  though  not  the  only  correct  way.      In  a  variation,  for  instance,  of  15 

to  20,  the  variation  element  is  a  variation  of   ]   to  1|,  or  a  variation  by  |. 

But  the  nominally  single  variation  of  15  to  20  differs  from  the  variation  of  1 

20       1^  X  15 
to  1^  in  that  it  contains  fifteen  such  variations  ;  for  ".  =  -i — -,.   ,    and    we 

lo         1  X  l-> 

have  a  variation,  not  of  1  l)y  |,  but  of  15  by  |.  Now  in  a  comparison  of  the 
arithmetic  averages  at  the  two  periods  (with  weighting  according  to  the  num- 
bers of  figures  reported  in  the  classes,  their  reported  sizes  also  being  used)  this 
difference  between  tlie  variation  of  15  to  20  and  the  variation  of  1  to  1^ — 
these  figures  being  supposed  to  appear  in  two  otherwise  similar  sets — shows 
itself  by  the  fact  that  a  different  result  is  obtained  according  as  we  use  the  one 
or  the  other  of  these  sets,  although  they  contain  the  same  variation  elements. 
But  in  the  arithmetic  average  of  the  variations  (with  weighting  likewise  ac- 
cording to  the  numbers  of  figures  and  variations  reported  in  the  classes)  this 
difl'erence  is  not  allowed  for,  the  variation  of  15  to  20  and  tlie  variation  of  1 
to  1;^  having  exactly  the  same  influence  upon  the  result. 

The  former  may,  then,  be  the  truer  expression  even  for  the  average  of  the 
variations — and  the  average  of  the  variations  must  be  adapted  to  it  by  using 
a  difTerent  weighting— in  all  those  problems  in  which  we  wish  the  variation 
to  count  according  to  the  sizes  of  the  figures  ;  '^  but  not  otherwise. 

-  In  general  we  want  the  size  factor  in  the  weighting  of  variations  in  all  those 
cases  in  which  greater  effort  is  needed  to  produce  the  same  variation  in  a  greater 
than  in  a  smaller  quantity.  Also  the  average  of  the  variations  with  weighting 
according  to  tlie  numbers  actually  reported  in  the  classes  can  obviously  be  correct 
only  in  those  problems  in  w  hich  it  is  possible  to  state  the  figures  only  in  one  way  ; 
for  otherwise  the  average  of  the  variations  would  depend  upon  the  accidental  way 
in  whicli  tlie  figures  that  vary  happen  to  be  reported.  It  may  happen  even  that 
we  wish  the  variations   to  count  inversely  according  to  the  sizes  of  the  figures. 

Then  the  weisrhts  of  the  variations  would  be       ,  ,    , ,  and  the  formula, 


V  = 


0.2 

+ 

62 

+ 

1 

1 

•  + 

b, 

+ 

«i 

Or  if  iti  any  cases  the  weighting  should  be  inversely  according  to  the  sizes  of  the 
classes,  viz.,  ,     r- , ,  the  formula  would  be 

JL2_  _,.  Jtl_  ^ 


1  1 


506  APPENDIX    A 

We  have  discovered  two  factors  in  tlie  weighting  of  the  variations — the 
numbers  of  the  figures  eniph)yed,  and  tlieir  sizes  at  the  first  period.  We 
have  discovered  eacli  of  these  upon  eliminating  the  other.  We  must  now 
unite  the  two.  As  they  both  act  directly,  their  influences  work  together  and 
strengthen  each  other.     Therefore,  in  all  cases, 

§  8.  The  variation  of  the  arithmetic  averages  of  classes  of  figures,  each  of  these 
averages  with  weighting  according  to  the  numbers  of  figures  in  the  dasses,  is  the  same 
as  the  arithmetic  average  of  the  variations  with  weighting  according  to  the  sizes  of  the 
classes  at  the  first  period. 

The  expression  for  the  average  of  the  variations  with  weighting  according 

On       I  J       On       , 

m,      +2/0,  ,    + ,     ,     , 

,  .  a,  b,  ,  •  1        ,  ^^9  +  '/Oj  + 

toxct,,  wo,,  ,  IS ,-  , — , ,  which  reduces  to  — ^   ,-^v— , , 

'  mi  +  2/6i+ '  xaj  +  2/6i+ ' 

which  is  the  expression  for  the  variation  of  the  averages.     Q.  E.  D. 

In  any  such  expression  as  the  last  we  should  notice  that  the  mathematical 

terms  are  not  X  and  y,  ,  Oj  and  h^,  •,  but  xa^,  yb^, Therefore 

in  the  denominator  the  one  term  xa^  may  be  replaced  by  ai,  the  one  term  yb^ 

by  b,,  and  so  on.     Then  in  the  numerator  the  terms  become  ai    ^ ,  b,  ,  '^ ,  and  so 

a,         0, 

on.     The  expression  then  falls  under  §  7. 

It  may  be  remarked  that  the  method  of  averaging  the  variations  has 
this  advantage,  that  it  tells  us  what  we  are  doing,  while  the  method  of  com- 
paring the  averages  hides  this.  We  have  now  discovered  that  when  we  com- 
pare the  arithmetic  averages  of  the  figures  which  have  varied  and  of  the 
figures  to  which  these  have  varied,  we  are  virtually  averaging  the  variations 
with  weighting  according  to  the  total  sizes  of  the  classes  at  the  first  period. 
Hence  we  may  view  this  weighting  of  the  variations  as  hidden  or  latent  in 
the  method  of  comparing  the  averages. 

From  this  follows  a  simple  corollary  : 

^9.  If  the  sizes  of  the  classes  at  the  first  period  are  all  equal  (i.  e., 
xoj  ^yb^  =^ ),  the  vari/dion  of  the  arithmetic  averages,  each  with  weighting  ac- 
cording to  the  numbers  of  figures  in  the  classes,  is  the  same  as  the  arithmetic  average 
of  the  variations  with  even  iveighting. 

This  condition  is  brought  about  when  the  numbers  of  the  figures  in  tlie 
classes  are  in  inverse  proportion  to  the  sizes  of  the  figures  at  the  first  period. 

III.    HARMONIC  AVERAGING. 
'i  1.   Witii  single  figures  in  the  two  sets,  the  liarmoiiic  average  at  the  first 
l)erio(l  is  ,    - , 5 t  ;  and  that  at  the  second  is  like  unto  it.     There- 

;(..,+»;  + ) 

fore  the  variation  of  the  averages  is 

1 

Ai  _  n  V  a.,  ^  6.,  ^  ) 

A   ~  1 


wlik'li  reduces  to 


and  to 


VARIATIONS    AND    AVERAGES  507 


A.   V'  +  V V 


1  +  1  + 

Ai      1  +  1  +  ..,...' 

a,  ^  6,  ^ 

(wliicli  sliows  that  it  is  indifferent,  in  comparing  liarnionic  averages,  whether 
we  inversely  compare  tlie  arithmetic  averages  of  the  reciprocals  of  the  figures 
or  the  sums  of  the  reciprocals  of  the  figures). 

Here  tlie  weighting  of  the  figures  in  eacli  average  is  even,  each  figure 
counting  once. 

§  2.  Given  the  same  single  figures,  the  liarmonic  average  of  the  varia- 
tions, likewise  with  even  weighting,  each  variation  counting  once,  is 

1 


A":    t       ) 


which  reduces  to 


1  /ttj       ftj 


(?,  +  h ) 


This  is  a  different  expression,  not  universally  reducing  to  the  jireceding. 
§  3.   With  classes  of  figures,  the  harmonic  average  at  the  first  j>eriod  may 
be  expressed  in  full  thus, 

M  (^:"^'l+ to  .r  terms)  +  ( ;!^  +  ^^  + to  ,  terms  )  + 

to  n  classes  j- 

which  maybe  abbreviated  t<T  ,  or  may  also 


1    /•'','/    ,  \  " 

+  ;    +    to  «  terms 


be  written   '         ■'  .     And  the  harmonic  average  at  tlie  second  period 

is  like  unto  it.       Therefore  the  variation  of    tlie  averages,  after  reductions 
similar  to  those  used  in  §  1,  is 

■'■  +  ?/  + 


.1X2  _.  "1 

Ai      •'■   .   .'/ 


Here,  too,  tiie  weighting  of  the  figures  in  each  average  is  directly  accord- 
ing to  the  numbers  of  times  the  figures  are  repeated  in  each  class.  No  other 
wav  of  averaging  the  sets  would  average  the  figures  siii)iiosed. 


50S  APPENDIX    A 

?.  4.  (Jiveii  tlu-  same  classes  of  linuivs,  tlie  similarly  weighted  harmonic 
average  of  the  variations  in  fnll  is,  after  the  first  reductions  in  the  denomi- 
nators, 

V  = -  -' 

.H(::+;;;- "•""-)+(';-"6:+ '<•!"--)+ 

)(  classes  I- 


to 

wliieii  niav  l)e  al)l)reviated  to 

V  1 


1  /    a,  h,  \ 

Ax      +  V  ,    + to  n  terms  ) 

n'  \    a.,       '  b.,  ) 


or  may  also  be  written 

^J^^lAy:^ 

a"  -  +  y  i  + 

Again  this  is  a,  different  expression,  not  nniversally  redncing  to  the  pre- 
ceding. 

?  5.  Here  also  we  may  suhsnme  single  fignres  under  classes,  and  confine 
onr  attention  to  the  latter.     Thus, 

2'/(e  variation  of  the  harmonic  avrrai/es  and  the  harmonic  average  of  the  varia- 
tions, in  all  cases  with  weighting  according  to  the  numbers  of  figures  in  the  classes,  are 
not  universally  the  same. 

?  6.  The  variation  of  the  harmonic  averages  aiul  the  harmonic  average  of  the 
variations,  all  with  weighting  according  to  the  numbers  of  fignres  in  the  classes,  are 
the  same  when  the  figures  at  the  first  period  are  equal. 

Let  a,  ^=  !>i  = =s.     Then  the  expression  for  the  comparison  of  the 

averages  in   ''/.  3  reduces  to  '  ;  and  the  expression  for  the 


■{:<^ ) 


average  of  the  variations  in  i  4  reduces  to  the  same.     C^.  E.  D. 

Here  is  the  same  factor  affecting  the  variation  of  the  harmonic  averages  as 
we  found  in  the  case  of  the  variation  of  the  arithmetic  averages,  namely  the 
number  of  times  the  figures  are  repeated  in  each  class,  or  weighting  of  the 
variations  directly  according  to  the  numbers  of  the  figures. 

In  all  other  cases  (except  of  course  when  all  the  variations  ari'  equal — 
see  I.  ?C)')  the  average  of  the  variations  will  require  difierent  weighting 
of  tlie  variations  to  make  it  agree  with  the  variations  of  the  averages. 

?  7.  When  the  numbers  of  the  figures  in  the  cktsses  are  all  equal,  the  variation  of 
live  harmonic  averages  {each  with  weighting  according  to  the  numbers  of  figures  in  the 
classes — in  this  case  with  even  weighting)  is  the  same  as  tfie  harmonic  average  of  the 
variations  with  weighting  inversely  according  to  the  sizes  of  the'  figures  at  the  first 
period. 

Let  r  -=■  //  -= :^  t.     Then  the  expression  for  the  variation  of  tlie  aver- 

'  Here  also  let     "       ,^    ^ *■ ,  and  iritri)(liir(>  /•  in  tlie  formula  given  here 

«i       l>i 
in  i>  4,  and  it  reduces  to  r .     Then  (/_,      /v(,  ,  li.^      rli^  ,  and  so  on  ;  and  l)y  intro- 
dnein'4  thene  values  in  the  t'otnnda  in  is:'.,  it,  t(»o,  reduces  to  r . 


VAIMATIONS    AM)    A\Kl{A(;i;s  509 

/  1  1  \ 

ages  in  ?  8  reduces  to  '  '  ,  ami  this,  liy  dropiiins,'  tin-  /  from 


(:-';+ ) 


both  sides  of  the  tiaetion,  to  the  expression  for  tlie  variation  of  tlie  averages 

of  the  figures  taken  singly,  /.  e.,  with  even   weighting.     We  want  now  to 

prove  that  this  expression  is  the  same  as  tlie  expression  for  tlie  average  of  tlie 

.  ,        .   ,    .                 ,.            11 
variations  with  weighting  accordnig  to      ,  ,   , 

a,    /', 

The    expression    for    the    average    of     tlie    variations    so    weighted    is 

111  wliicli  /(  -j-       -t-  to  n 


l/lai,16i,  ^        ^  V 


terras  ;  wherefore  this  expression,  hv  redufinti:.  restoring  tlie  value  of  //'.  and 
converting:,  becomes  -  '        ,'  ,    wliicli    is   tlie   same  as   the  iin-ccding. 

'+    + 

Q.  E.  1). 

Thus  here,  too,  a  factor  which  affects  the  variation  of  the  averages  is  the 
size  of  the  figures  in  the  classes  ;  only  here  the  weighting  of  the  variations  is 
inversely  according  to  the  sizes  of  the  figures. 

This  is  the  opposite  of  what  was  the  case  in  the  variation  of  the  arith- 
metic averages.  There  the  larger  a  figure,  the  more  its  variation  counts. 
Here  the  larger  a  figure,  the  less  its  variation  counts. 

Of  course  in  the  harmonic  average,  as  in  all  averages,  the  larger  a  figure, 
the  larger  is  the  average  of  the  set  of  figures  in  Avliich  this  figure  occurs  ; 
and  similarly,  the  larger  a  variation,  the  larger  is  the  average  of  the  set  of 
variations  in  which  this  variation  occurs.  The  proposition  before  us  is  that 
the  larger  the  figure  which  varies,  the  smaller  is  the  infiuence  of  its  varia- 
tion upon  the  variation  of  the  averages.  The  reason  of  this  is  because  the 
harmonic  average  is  the  reciprocal  of  the  arithmetic  average  of  the  recip- 
rocals ;  but  the  larger  a  figure,  the  smaller  is  its  reciprocid. 

We  have,  then,  discovered  two  factors  in  the  weighting  of  the  variations 
— the  numbers  of  figures,  and  their  sizes  at  the  first  period.  But  as  the  one 
of  these  acts  directly  and  the  other  inversely,  their  combined  influence  is  the 
balance  left  over  as  the  one  neutralizes  the  other.  Therefore,  taking  l)oth 
into  account,  in  all  cases, 

§  8.  The  variation  of  the  harmonic  avera(/es  0/  c/a.ssfs  of  figures,  eacli  of  these 
averages  with  weighting  directly  according  to  the  numbers  of  the  figures  in  the  classes, 
is  the  same  as  the  harmonic  average  of  the  variations  with  weights  which  are  the  ratios 
of  the  numbers  of  figures  in  the  classes  to  tfte  sizes  of  these  figures  at  the  first  period. 

The  expression  for  the  average  of  the  variations  with  weighting  according 

•'■+^  +  ■'■4-^+-  

X      y  .  a.        t),  I  •   1         1  .     "1        "1 

to       ,  ■    ,  ,  "^  •  wliicti  icduces  to  , 

«!      f>l  •'■    .  "1  _L  •'/    .  ''1  4. ■'     4-    •'    -f 

f(i    ((2        l>i    b.,  a..       b., 

which  is  the  expression  for  the  variation  of  the  averages.     (^.  K.  I). 
From  this  also  follows  the  simple  corollary  : 


510  AIM'KNDIX     A 

§  0.   If  the  ratios  of  the  numbers  of  fi(/ures  in  the  classes  to  the  .s/.-e,s  of  the  Jif/ures  at 

the  first  period  are  all  equal  (  i.  e.       z=  ^  = j  ,  the  variation  of  the  harmonic 

averages,  each  with  weightinfj  according  to  the  numbers  of  figures  in  the  classes,  is  the 
same  as  the  harmonic  average  of  the  variations  with  even  weighting. 

This  condition  is  brought  about  when  the  numbers  of  the  figures  in  the 
classes  are  in  direct  proportion  to  the  sizes  of  the  figures  at  the  first  period. 

§  10.   A  remark  deserves  to  be  added. 

We  liave  seen  that  in  the  aritlimetic  averaging  the  expression  for  the  varia- 
tion of  the  averages  Avas  in  some  cases  truer  tlian  the  expression  for  tlie  aver- 
age of  the  variations  (unless  this  has  its  weighting  specially  adapted),  because 
it  gives  weight  to  the  figures  according  to  their  numbers  and  sizes  in  the  de- 
nominator. Now  in  the  same  case  we  might  here  still  want  the  figures  to 
count  directly  according  to  their  sizes  as  well  as  directly  according  to  their 
numbers — we  might  want  the  variation  of  a  larger  number  to  count  more  in- 
stead of  less.  Therefore  the  comparison  of  the  harmonic  averages  would  not 
be  serviceable  for  this  purpose,  and  in  order  to  carry  it  out  we  nuist  insert  the 
desired  weighting  in  the  expression  for  the  harmonic  average  of  the  varia- 
tions, thus, 

V 


1    /       "l    ,      ,    I'-,    , 
—;,  I  ''"i      +  '/'Ji  ,    +  tf)  n  terms 


which  reduces  to 

_  _av(i  -f-  //6,  4- 

^      O  2  7,  2  ' 

.r"^+y7   + 

cr.,  I'., 

or,  if  we  represent  xa^  by  a,  yb^  by  b,  and  so  on,  (or  supposing  the  weights 

are  sometimes  wanted  to  be  something  else,  still  letting  a  and  b  repre- 

.sent  them)  we  should  have 

1 


-^  (  a    '  +  b  , '  -f to  n  terms 


in  which  Ji''-=a  +  b+ to  ;;  terms.     The  expression  for  the  variation 

of  the  averages,  however,  is  serviceable  on  condition  that  the  figures  at  the 
firet  period  are  reduced  to  equality  ;  for  then  the  numbers  of  times  they  are 
repeated  in  their  classes  are  proportional  to  a,  b, 

IV.    CASES  OF  AGREEMENT   BETWEEN  THE   ARITHMETIC  AND 
THE  HARMONIC  AVERAGES  OF  VARIATIONS. 

We  have  noticed  in  the  comparison  of  the  arithmetic  averages  what  hap- 
pens when  f(i  =  6j  = ,  and  when  xa^  ^yb^  ^= ,  and  in  thecomparison 

of  the  harmonic  averages  what  happens  also  when  ai-=b^^^ ,  and  when 

--^  ,  -- There  remains  to  see  what  happens,  in  the  former  case,  when 

«,       6i 

«j  =:  6j  = ,    and    when    i,a^z:^yh^^=^ ,    and    in    the    latter,    when 

0   =:  6,    -  ,  ;ind   when        =^      r=: A  few  other  coincidences  will 

'(..      a, 
also  call  f(ir  altciitiou. 


VARIATIONS    AND    AVEI{A(;ES  oH 

^  1.   The  variation  of  tlie  aritlimetic  avcrafjes  being  expressed  tiius, 

Aj  ^jrOj  +  .'A-'H — -^ 
A|      j-o,  +  //A,  + ' 

we  know  tliis  to  l)e  tlie  same  as  tlie  arithemetic  average  of  tiie  variations  witii 
weighting  according  to /«,,  yb^, 

Letrf2^=62= ^s.     Then  .ra,    =  j-rtj    '  =  Js    ' ,   and  simihirly   yh^ 

h  x^  "4"  vs  "T"  ■■*•*• 

I/.-!    ' ,  and  so  on.     Tlierefore  tiiis  expression  becomes 7 , 

^■%2-^^^*2    + 

Qf*  —I—  y  — 1~      

which  reduces  to     — — --r  ,  which  we  recognize  as  the  expression 

':;+■■'::+ ....'. 

for   the   harmonic  average    of   the    variations    witli  weigliting  according  to 

X,  y, Therefore, 

If  it  happens  that  the  figures  at  the  second  period  are  all  eqwd,  the  arithmetic 
average  of  the  variations  with  weighting  according  to  the  sizes  of  the  classes  at  the  first 
period  is  the  same  as  the  harmonic  average  of  the  variations  ivith  weighting  according 
to  the  numbers  of  figures  in  the  classes. 

§  2.  Let  rrtj  =^  2/^2^ =  '•  Then  a-tti^/rtg   '  =  <    '  >  ii"d  similarly  yb^  = 

a^         ^2 

t  r  ,  and  so  on.    Therefore  the  above  expression  becomes 1 , 

'^  t"^^tl^-\- 

a.,  0., 

which  reduces  to  z.  — , r- ,  which  we  recognize  as  the  expression 


'h+1+ ) 

n  xa^       02  ) 


for  the  harmonic  average  of  the  variations  with  even  weighting.     Therefore, 
If  it  happens  that  the  sizes  of  the  classes  at  the  second  period  are  all  equal,  the 

arithmetic  average  of  the  variations  with  weighting  according  to  the  sizes  of  the  classes 

at  the  first  period  is  the  same  as  the  harmonic  average  of  the  variations  with  even 

weighting. 

This  theorem  may  be  extended,  as  follows  : 

^  3.     The  harmonic  average  of  the  variations  with  weighting  according  to 

the  numbers  of  figures  in  the  classes  being  this,    ~ ,    suppose 

^1    r          1    I 
•V  +  ^^,+ 

that  instead  of  this  weighting  we  use  weigliting  according  to  the  sizes  of  the 

classes  at  the  second  })eriod.     Then  we  mnst  substitute  .raj  for  .r,  yb.,  for//,  and 

,         ,  xa.,  -\-  ijb.^  -\- ...        ,  xa.,  +  yb.,  -V-     — 

so  on,  and  we  have  "       '   -. ,  wlucli  reduces  to      '  ,  -^—  , 

,„„".  +  '^^^- ■'■"i+.'A+ 

■■  a.,       '    "  li., 
which  we  recognize  as  the  expression  for  the  variation  of  the  aritmetic  aver- 
ages with  weighting  according  to  x,  y,  ,  and  which  we  know  to  be  the  same 

as    the   arithmetic   average   of  the   variations  with   weighting   according    to 

j;a,,  2/6j Therefore,  in  all  cases, 

The  arithmetic  average  of  any  varifitions  villi  weighting  according  to  the  sizes  of  the 
classes  at  the  first  period  is  the  same  as  the  hfumonic  average  of  the  variations  unth 
weighting  accordin<i  to  the  sizes  of  the  classes  at  the  second  period. 


512  APPENDIX    A 

§4.   The  variation  of  the  harnioiiic  averages  being  expressed  thus, 

^+  !'  -^ 

A2  _  ffli     ^1 

a,  "^  I,  ^ 
we  know  this  to  be  tiie  same  as  the  harnionie  average  of  the  variations  with 

...  1-  1  .  X         1/ 

weighting  according  to  the  ratios       ,  ;    , 

^         ^  *  f(,     />, 

"1        ,  •'■        -^i  1    •     -1     1     '/ 

Leta.,  =  6.,= =s.      llien((,=.s      and       =-      " ,  and  siimlarlv  !     ;= 

a.,  «,        sa^  0, 

,  r     1.      r 

^ ,  and  so  on.     Therefore  this  expression  becomes  — ,  wliich 

A'  -+^  + 

reduces  to       '    , ] ,  wiiich  we  recognize  as  the  exi>ression  for  tiie 

•'-l-y+ 

arithmetic  average  of  the  variations  with  weighting  according  tn  /.  //, 

Therefore, 

If  it  luippens  that  the  figures  at  the  second  period  are  all  equal,  the  harmonic  aver- 
age of  the  variations  with  weighting  according  to  the  ratios  of  the  numbers  of  figures  in 
the  classes  to  the  sizes  of  the  figures  at  the  first  period  is  the  same  as  the  arithmetic 
average  of  the  variations  with  weighting  according  to  the  numbers  rf  figures  in  the 
classes. 

S  5.   Let   '    -=  r^  =  ^=  r .      Then    r— m.,,   and    -  ==  —   ;    and   sinii- 

a.,        62  ^'1         ''1 

larlv  .    =^   ,  '  ,     and    so    on.      Therefore     the    above    expression     be<-omes 

«2     I         *2     1 

^      -T->'r  + \  /  (I        l>  \ 

— 3— i ,  wiiicii  reduces  to  -  (     " '+  ,"  + )  ,  which  we  recog- 

)•  +  ;•+ n  \«i       hi  ) 

nize  as  the  expression  for  the  arithmetic  average   of  the  variations  with  even 

weighting.     Tlierefore, 

If  it  happens  that  live  ratv^s  of  the  numbers  of  figures  in  the  chtsses  to  the  sizes  of 
the  figures  at  the  second  period  are  all  equcd,  the  harmonic  average  of  the  varia- 
tions with  iveighting  according  to  the  ratios  of  these  numbers  to  the  sizes  of  tite  figures 
at  tJie  first  period  is  the  same  as  the  arithmetic  average  of  the  7'(iriations  with  even 
weighting. 

Tiiis  theorem  also  may  be  extended,  as  follows  : 

§  ().   Tlie  aritlimetic  average  of  the  variations  with  weighting  according  to 

"2    ,       h    , 
tlu^   iuniil)ers   of   ligures  in   tlie    classes  being  this,    — -  , '  ,  sup- 

^'-\-y-r 

pose  that  instead  of  this  weighting  we  use  weighting  according  to  the  ratios 
of  the  numbers  of  figures  in  the  classes  to  the  sizes  of  the  figures  at  the  sec- 
ond period.     Then  we  must  substitute      for.r    ;-  for  (/,  and  soon,  and  we  have 

'(.,  h., 


VAIIIATIONS    AND    AVERAGES  513 

•^    /i?  4.  1    .   ^2  _|_ Ia.1^ 

tu,     a,       b„     b,  ,  .  ,        ,  «i       ^1  ,  .  , 

-^ ' ' * ,  which  reduces  to ,  which  we  recog- 

a^  ^  62  ^                                                       a.,  ^  b,  ^ 
nize  as  the  expression  for  the  variation  of  the  harmonic  averages  with  weight- 
ing according  to  x,  y ,  ,  and  which  we  know  to  be  the  same  as  the  har- 
monic average  of  the  variations  with  weighting  according  to      ,  f  , 

"a,    Oj 

Therefore,  in  all  cases, 

The  harmonic  average  of  the  variations  with  weighting  according  to  the  ratios  of 
the  numbers  of  figures  in  the  classes  to  the  sizes  of  the  figures  at  the  first  period  is  the 
same  as  the  arithmetic  average  of  the  variations  with  iveighting  according  to  the  ratios 
of  these  numbers  to  the  sizes  of  the  figures  at  the  second  period. 

^  7.  Suppose  the  numbers  equal  the  variations. 

Then  x  =  —  ,  2/  =  5^ ,  and  so  on  ;  and  the  harmonic  average  of  the  varia- 

''^  +  'i+ 

tions  with  weighting  according  to  .r,  )/,  becomes  ^ r^— r , 

^.^_|_^.^_L 

which  reduces  to      I  —  +  7-  + ) ,  which  we  recognize  as  the  arithmetic 

n  \a^       b^  ) 

average  of  the  variations  with  even  weighting.     Therefore, 

If  it  happens  that  the  number  of  figures  in  every  class  equals  the  variation,  the 

arithmetic  average  of  the  variations  with  even  weighting  is  the  same  as  the  harmonic 

average  of  the  variations  with  weighting  according  to  the  numbers. 

§  8.  Suppose  the  numbers  equal  the  reciprocals  of  the  variations. 

Then  x  =    - ,  y  =  j/  ,  and  so  on  ;  and  the  arithmetic  average  of  the  varia- 

^h.^hih.b^ 

tions  with  weighting  according  to  X,  y,  becomes  r , 

a.'^b,'^ 
which  reduces  to  :;-— -. r- ,  which  we  recognize  as  the  harmonic 


-(-  +  ?'+      ) 

n  \  Oj       ('2  / 


average  of  the  variations  with  even  weighting.     Therefore, 

If  it  lutppens  that  the  number  of  figures  in  every  class  equals  the  reciprncal  of  the 
variation,  the  harmonic  average  of  the  variations  with  even  iveighting  is  the  same  as 
the  arithmetic  average  of  the  variations  uith  weighting  accoi'ding  to  the  numbers.^ 

^  Although  aside  from  our  subject,  yet  as  throwing  light  upon  it,  the  follow- 
ing two  theorems  may  be  added  : 

The  simple  arithmetic  average  of  any  quantities  is  the  harmonic  average  of 
them  ivith  weighting  directly  according  to  their  sizes. 

Of  the  quantities  a,  b,  ,  k,  the  harmonic  average  with  weights  di- 
rectly  according   to    their   sizes   is    — .-  ,    which  reduces   to 

a-  +  b^+ +*T 

a  b  k 

—  (a  +  6  + +  k),  which  is  their  simple  arithmetic  average.    Q.  E.  D. 

n 

33 


514  APPENDIX    A 

V.   GEOMETPJC  AVERAGING. 
§  1.   With  single  figures  in  the  two  sets,  the  geometric  average  at  the  first 

period  is  {/""(ti  ■  b^ to  n  terms,  and  that  at  the  second  is  like  unto  it. 

Therefore  the  variation  of  the  averages  is 

A2_|X«2-&2 

Ai       v'^ai-bi 

(Here  it  may  incidentally  be  remarked  that  the  comparison  of  these  averages 
cannot  be  replaced  by  comparison  of  the  products  except  when  the  result  is 
unity  ;  for  the  products  are  in  a  ratio  the  nth  power  of  the  ratio  between  the 
averages. ) 

,  Here  the  weighting  of  the  figures  in  each  average  is  even,  each  figure 
counting  once. 

^  2.  Given  the  same  single  figures,  the  geometric  average  of  the  variations, 
likewise  with  even  weighting,  each  variation  counting  once,  is 

This  is  an  expression  which  universally  has  the  same  value  as  the  pre- 
ceding. 

§  3.  With  classes  of  figures,  the  geometric  average  at  the  first  period  may 
be  expressed  in  full  thus, 

Ai  =  t/'C"!  ■«! to.r  terms)  •  [b^-bi toy  terms) to  n  classes, 

which  may  be  abbreviated  to  y  a,^  •  ^i" to  n  terms.     And,  the  average 

at  the  second  period  being  similar,  the  variation  of  the  averages  is 

A2 l/o-2''bj' to  n  terms 

A,       V  (h^  •  ^i" to  n  terms 

Here  the  weighting  of  the  figures  in  each  average  is  directly  according  to 
the  numbers  of  times  the  figures  occur  in  each  class.  No  other  way  of  aver- 
aging the  sets  would  average  tiie  figures  supposed. 

I  4.  Given  the  same  chisses  of  figures,  the  similarly  weighted  geometric 
average  of  the  variations  in  full  is 

The  simple  harmonic  average  of  any  qtiantities  is  the  arithmetic  average  of 
them  with  weighting  inversely  according  to  their  sizes. 

Let  k  be  the  largest  of  these  quantities.     Then  the  weights  of  tliese  quantities 

according  to  the  inverse  of  their  sizes  are        for  a ,  -    for  b, ,  and   ",    (or  1 ) 

a.  b  k 

for  k  .    T!ie  arithmetic  average  of  these  quantities  so  weighted  is 

1 Ik        _^  k     ,   ^  ^  ,     /'M 

a         ij  k 

wliich  reduces  to wliicli  is  their  simple  harmonic  aver- 


n  ^  a       0  k  I 

age.     Q.  E.  D. 


VARIATIONS    AND    AVERACiES  515 


n'/  (iu    cu                                     \      (b~    ft,                                     \ 
=  I /     I       •    " to  .);  terms  I  •  I  ,    •  , " to  »/  terms  ) 

to  n  classes, 


V 

wliii'h  mnv  be  ab])reviate<l  to 


^=i^(^;)'-(t:)' ™- 

Again  this  is  an  expression  whicii  universally  has  the  same  value  as  the 
preceding. 

\  5.   Here  also  subsuming  single  figures  under  classes,  we  see  that 

The  variation  of  the  geometric  averages  and  the  geometric  average  of  the  variations, 
in  all  cases  with  weighting  according  to  the  numbers  of  figures  in  the  classes,  are  the 
same. 

We  see  also  that  this  agreement  will  universally  take  place  with  any 
weighting  whatsoever,  provided  it  be  the  same  in  all  the  three  avei'agings. 

Thus,  unlike  the  other  two  kinds  of  averaging,  in  the  geometric  averaging 
for  this  agreement  to  take  place  it  is  not  necessary  that  a^  =;  ij  = ,  or 

that  ^  =  7~= '^^  since  the  agreement  takes  place  not  only  in  these  but 

in  all  cases. 

§  6.  If  we  employed  here  the  weighting  which  we  found  necessary,  except 
in  the  above  two  cases,  for  the  arithmetic  average  of  tlie  variations  in  order 
that  it  should  agree  with  the  variation  of  tiie  arithmetic  averages,  namely, 
according  to  xa^,  yb^, ,  which  would  be  in  this  form. 


i^C;)""  •(:•;)""• - 


n  terms, 


in  which  n'^^^xa^  -|-2/^i  + to  w  terms  ;  or  if  we  employed  here  the  simi- 
larly necessary  weighting  in  the  harmonic  averaging,  namely—,   V- ,  , 

a,     Oi 

which  would  be  in  this  form, 


V         [i)     \i)    ■ tonyerm., 

in  which  n"  —  -   4-  ;    4-    to  n  terms  ;  either  of  these  expressions  for  tlie 

average  of  the  variations  would  agree  with  the  expression  for  the  variation 
of  the  averages  (apart  from  the  case  when  all  the  variations  are  alike)  only 
in  a  particular  case,  namely  if  the  figures  are  all  the  same  at  the  first  period 

(i.  €.,«!  =  6i=   ) — the  very   condition    previously  found   necessary   to 

make  averages  of  the  variations  with  weighting  like  the  weighting  in  the 
separate  averages  of  the  figures  agree  with  the  variations  of  these  averages. 

^  That  in  this  particular  case  the  results  are  always  the  same  as  any  one  of  the 
variations,  whatever  be  the  weighting  employed  (if  only  it  be  the  same  in  the 
two  averages  of  the  figures  that  are  compared),  may  easily  be  seen  by  letting 

—  =  T^  =   =  r,  and  by  introducing  r  into  the  above  expressions  with  any 

ax       Oi  — ~ 

weights.     These  expressions  then  always  reduce  to  r. 


516  APPENDIX   A 

Tlierefore  if  here,  with  uneven  vari;itions,  the  figures  being  unequal  at  the 
first  period,  we  reduce  these  to  equality  and  adapt  their  numbers  accordingly, 
and  employ  weighting  according  to  the  adapted  numbers  of  figures  in  the  classes, 
we  shall  get  a  result  for  the  average  of  the  variations  thus  weighted  (really 
according  to  Ta-^,  yb^,  )  different  from  that  of  the  variation  of  the  aver- 
ages (in  which  the  weighting  is  according  to  :r,  .)/,  ). 

§  7.  The  reason  for  the  universal  agreement  in  this  case  is  that  liere  the 
sizes  of  the  figures  at  the  first  period,  in  the  comparison  of  the  averages,  is 
offset  by  their  sizes  at  the  second  period,  and  do  not  affect  the  result.  Hence 
there  is  here  no  weighting  in  the  comparison  of  the  averages  except  the  num- 
bers of  times  the  figures  occur  in  the  classes.     There  is  no  hidden  weighting. 

Therefore  if  we  wish  to  employ  weighting  which  also  allows  for  the  sizes 
of  the  figures  at  the  first  period  (or  at  any  other  period),  we  have  to  intro- 
duce this  into  the  separate  averages  which  we  compare  as  well  as  into  the 
averages  of  the  variations. 

VI.   COMPARISON  OF  THE  GEOMETRIC  AVERAGE 
AVITII   THE   OTHER  TAVO. 

§  1.  Of  the  same  given  numbers  of  the  same  given  figures,  at  least  one  of 
which  is  nne(iual  to  the  others,  the  arithmetic  average  is  always  greater  than 
the  harmonic.     Thus 


xa-i-yb  -ir -^  a^  +  2/ + 


a^b+ 

The  demonstration  of  this,  which  involves  the  demonstration  that 

{.ra  +  yb+ )('7+f+ )>(■'•  + i/  + )S 

is  somewhat  elaborate,  and  need  not  be  given  here. 

This  being  true  when  the  terms  are  figures  (or  integers),  it  is  also  true 
when  the  terms  express  the  variations  of  figures  (or  are  fractions).     Hence 

With  the  same  weighting,  the  anthmetic  average  of  the  same  variations  is  always 
greater  ( or  higher )  than  the  harmonic.  That  is,  when  the  averages  are  above 
unity,  the  arithmetic  average  is  a  greater  variation  than  the  harmonic  ;  and 
when  the  averages  are  below  unity,  the  arithmetic  is  a  smaller  variation 
than  the  harmonic  ;  or  the  arithmetic  may  be  above  while  the  harmonic  is 
at  unity,  or  may  be  at  unity  while  the  harmonic  is  below,  or  may  be  above 
while  the  liarmonic  is  below. 

§  2.  Given  only  two  figures  (or  two  figures  each  rej^eated  thesaujc  number 

of  times,  so  that  we  have  even  weighting),  the  arithmetic  mean  being      «      ' 

and  lliu  liarmonic  , r  =  — r-f  <  the  geometric  mean  between  these  means  is 

1,1        a-\-  b' 


xl 


a      b 


a  4-  b     2al)  /— Ti.i.i  •  i  i        •         n 

,    ,  ;  --  V  (lb  ,  whicli  IS  the  geometric  mean  between  tlie  given  ng- 
'1         a  -\-  b 

ures.     The  same  holds  good  if  the  figures  are  fractions  (representing  varia- 
tions of  figures).     Therefore 


VARIATIONS    AM)    AVERAGES  517 

Between  two  fiyurcn  {ar  raridlldiis),  each  taken  sirujly  {or  in  erjual  vumhent),  the 
geometnc  mean  is  the  (jeomctrie  mean  betirecn  the  arithmetic  and  tlic  harmonic  ineavf. 
Hence  also,  in  these  cases, 

The  geometric  mean  is  smaller  {lower)  than  the  arithvietic,  and  (/reater  {higher) 
than  the  harmonic. 

§3.   With  three  figures  eacli  taken  singly  (or  in  equal  numbers),  or  with 

two  figures  taken  uneven  times  (so  tliat  we  must  employ  uneven  weighting), 

the  first  of  these  propositions  does  not  hold  good  of  the  averages  between  them. 

For  between  three  given  figures,   each  taken  singly,   the  arithmetic  aver- 

.    a-{-  b  4-  c  ,    ,      ,  .  i^  oabc  ,    , 

age  IS  ^ ,   and  the  harmonic,  t, -=    ,    t^, ;   and  tlie 

®  3        '  '   1        1        1       ab-\-bc  +  ca  ' 

„     IT'     _' 


,                  ,                                     .       la  4-  ^  +  c 
geometric  mean  between  these  two  averages  is -vl s -^ 


3abc 


b  -{-  be  -\-  ca 


■\  abci  ~ — — , — , I  =      I- , ,  which  does  not  reduce  to  the  geo- 

V        \nb  +  bc-\-  ca/        ^  1    ,    1    i    1 

a        b        c 

metric  average  between  the  three  figures,    which  is  i/  abc.     Or  with  two 

figures  of  which  the  one  occurs  x  times,  and  the  other  y  times,  the  arith- 

.     xa4-yb           ,      ,       ,             .        r-\-y        {x-\-y)ab  , 

metic  average  is t-    - .    and    the    harmonic, = ; — r  ;    and 

X  + 1/  '     X        y         ya  +  xb 

a  "^  Z 


,  .  ,  ,  .        I  xa -\- yb    {x  +  y)ab 

the  geometric  mean  between  tliese  two  averages  is  -v   — • , — r-  ^ 

^   i+y      ya  +  xo 

■\  ab  I  — -p^  I  =      f —,  which  does  not  reduce  to  the  geometric  aver- 

V        \ya-\-xbJ        \  ^,y 

^  a^  b  

age  between  the  two  uneven  classes,  which  is  ^  ^  a'bv.      In  general,  a  set  of 

uneven  classes  of  figures  or  variations  may  be  analyzed  into  a  larger  number 

of   even   classes,    some    being    homonymous.     We   may,    therefore,    confine 

our   attention  to  sets  of  even   classes,    or   of  single   figures   or   variations. 

Of  a  number,   n,   of  such  figures,   the   geometric   mean   between  the  arith- 

1.1     1  •  •        f«  +  i'  + Ga- 

metic and  tlie  harmonic  means  is 


i 


i 


1  4.1+1 


,   which  does  not  reduce  to  the  geometric  average  of 


abc 


the  given  figures,  which  \s  {/^ abc ,  (provided  the  figures,  or  variations, 

do  not  constitute,  or  reduce  to,  two  classes  with  even  weighting). 

But  in  these  cases  the  second  of  the  above  propositions  still  holds,  and  we 
may  also  specify  it  more  definitely,  though  without  exact  demonstration. 
Thus  we  have  the  two  following  propositions,  the  first  of  which  is  demon- 
strable. 

§  4.  Of  any  figures,  or  variations,  with  any  weighting,  the  same  in  all  the  aver- 
agings, the  geometric  average  is  smaller  {lower)  than  the  arithmetic,  and  greater 
{higher)  than  the  harmonic. 

The  demonstration  of  the  first  part  of  this  proposition,  at  least  for  cases 


518  APPENDIX    A 

with  even  weiijlitin.a;  (whence  it  can  easily  be  extended)  is  generally  given  in 
works  on  algebra, '  although  notice  of  the  second  is  generally  neglected.  Of 
both  a  brief  indication  is  given  by  Walras  (B.  61,  p.  15).  In  full  tlie  demon- 
stration is  too  long  to  give  here. 

§  5.  Of  any  figures,  or  varialions,  with  any  weighting,  the  same  in  all  the  averag- 
ings {provided  it  be  not  even  weighting  for  only  two  classes  of  distinct  figures  or  vari- 
ations), the  geometric  average  is  sometimes  above,  and  sometimes  below,  the  geometric 
mean  between  their  arithnetic  and  hai-monic  averages. 

This  is  shown  by  trial.  But  it  is  evident  that,  if  different  at  all,  the  geo- 
metric average  must  vary  on  both  sides  of  the  geometric  mean  in  question  ;  for 
if  in  any  one  set  of  figures  or  variations  it  be  above,  it  must  be  below  in  a  set 
of  their  reciprocals  (,e.  g.,  it  is  above  in  2,  3,  4,  5,  and  below  in  ^,  |,  |,  |). 

The  easiest  way  to  make  the  comparison  is  as  follows.  Place  the  expressions 
for  this  average  and  for  this  mean  side  by  side,  the  former  on  the  left  and  the 
latter  on  the  right,  thus, 

{/abT^^^-  ,  r  '^l^    ; 

Ni+^+^+ 

a      0       c 
scjuare  each,  and  raise  each  the  n*  power,  thus, 

a-\-b-\-c-\- 


[abc y 


(r^+ r 


multiply  each  by  the  denominator  of  the  one  on  the  right,  thus, 

{'djc )'(J+  \^\^ )"'         ("  +  6-t-c  + )". 

The  side  on  which  superiority  appears  in  tlie  last  line  will  also  be  the  side  on 
which  it  exists  in  the  first  line.  But  this  method  does  not  exhi])it  the  pro- 
portionate amount  of  the  difference. 

Trial  seems  to  show  that  when  all  the  terms  are  above  unity,  or  when  the 
terms  in  the  numerators  (including  those  virtually  divided  by  unity)  sum  up 
to  a  greater  amount  than,  or  outweigh,  the  terms  in  tlie  denominators  { in- 
cluding the  omitted  units),  the  geometric  average  is  above  the  geometric 
mean  in  question  (as  in  \,  3,  \,  5);  and  if  reversely,  below  (as  in  2,  \,  4,  \'). 

In  intermediate  cases  the  geometric  average  may  coincide  with  the  geo- 
metric mean  between  the  other  averages. 

Trial  also  shows  that  the  difTerence  is  generally  very  slight,  although  it 
may  be  appreciable  if  only  a  few  of  the  figures  or  variations  are  very  differ- 
ent from  the  rest,  or  very  excessive  compared  with  the  average. 

\  6.  Now  of  given  numbers  of  given  figures  that  vary  between  two  periods, 
we  have  seen  that  the  arithmetic  average  of  their  variations  with  weighting 
according  to  the  sizes  at  the  first  period  is 

a.^   I      ,   b., 

Tax      -Vybi,    -\- ,      ,     , 

«i  ^  _xa,-^yb^-\- 


a;ai  +  2/^  + a-a^ -t- 2/6i -f- ' 

and  the  harmonic  average  of  their  variations  with  weighting  according  to  the 
sizes  of  the  classes  at  the  second  period  is 

^  E.  (J.  Tixllimitcr's  Algebra  for  the  use  of  schools  and  colleges,  §  (iSO. 


YARIATIOXS    AND    AVERAGES  519 

CTg-t-  yh^  -f _  xa^-{-y\A- 

-."-^  +  3/64^  xa.  +  Z/^ +  ••••••■ 

We  now  see  ( from  §  4 )  that  the  geometric  average  of  these  variations  with 
weighting  according  to  the  sizes  of  the  classes  at  the  first  period  is  smaller 
than  the  arithmetic  average  with  this  weighting,  thus, 

and  that  the  geometric  average  of  them  with  weighting  according  to  the  sizes 
of  the  classes  at  the  second  period  is  larger  than  the  harmonic  average  with 
this  weighting,  thus, 

xa^  +  yb„    /- 

in,y-.     lu^y.     ^ 

xtti  +  yb^  + 


+  -y'  n^Y"- ■  i^'Y"- ■ ^^«2  +  y^+_ 


And  we  see  (from  §  5)  that,  generally,  the  geometric  average  with  the  latter 
weighting  is  larger  nearly  in  the  same  proportion  as  with  the  former  weight- 
ing it  is  smaller,  than  this  one  and  the  same  average.  Therefore  if  we  take 
the  geometric  mean  between  the  weights  in  these  two  systems  of  weighting, 
and  employ  the  geometric  average  of  the  variations  with  weighting  according 
to  these  means,  thus, 

ry'a^„  +  y'^'bJl    /- ; — ■—=. — —— — — = 


1/  (I)  ■'■•(t:)' 


the  result  will  be  nearly  the  same  as  that  of  the  other  average.  But  because, 
in  §  5,  the  geometric  average  is  not  exactly  at  the  geometric  mean  between  the 
other  two,  but  inclines  to  the  one  side  or  to  the  other  according  as  the  classes 
preponderate  that  rise  or  fall,  so  the  geometric  average  with  this  intermediate 
weighting  will  incline  to  the  one  side  or  to  the  other  in  similar  cases. 

With  only  two  classes  with  even  weighting  the  geometric  average  is  exactly 
at  the  geometric  mean  between  the  other  two,  as  seen  in  §  2.  Hence  in  these 
cases  we  should  expect  the  following  proposition  to  be  demonstrable,  and,  in 
fact,  find  it  so. 

§  7.  0/"  the  variations  of  two  figures,  or  cl(tsses  of  figures,  such  that  the  products 
{and  hence  the  square  roots  of  these  products)  of  their  sizes  at  both  periods  are  equal, 
the  geometric  average  with  even  weighting  is  the  same  as  the  arithmetic  average  with 
weighting  according  to  the  sizes  at  the  first  period,  or  the  harmonic  average  ivith 
weighting  according  to  the  sizes  at  the  second  period. 

Treating  of  classes,  as  the  more  complex  case  including  the  other,  we  wish 
to  prove  that 

02    fcg  _  oca,  +  yb^ 
bi  "  -»«i  -f  yhi ' 

given  that  xa,  •  xn^  =  yb^  •  yb,  (this  condition  being  necessary  in  order  that  the 
weighting  of  the  geometric  average  may  be  even). 

yib  b. 

From  the  condition  we  obtain  02^-2-'^;  and  by  substituting  this  value 
of  a,  in  the  equation  to  be  proved,  we  have 


V; 


520 


which  reduces  to 


V 


APPENDIX    A 

y'^^^^+yb. 

\y'bA 

b,         xa^      '  y  ' 

iA_ 

yb.,{ybi  +  xa-i) 

a-ch 

.iY(^(.ra,  +  2/^1 )  ' 

which  is  evident ;  wherefore  the  first  equation  is  correct.     Q.  E.  D. 

With  only  two  figures,  that  is,  with  one  figure  in  each  class,  x  and  y  in  the 
above  are  units  (so  that  the  condition  is  0/(2  =  ^1^2))  ^"f^  the  result  works  out 
the  same. 

§  8.  Witli  three  or  more  figures,  or  classes  of  figures,  even  though  the  prod- 
ucts of  their  sizes  at  both  periods  be  all  equal,  the  statement  in  the  preceding 
proposition  does  not  universally  hold. 

It  holds  when,  there  being  no  variations  of  the  figures  in  the  other  classes, 
the  figures  in  two  such  classes,  or  in  any  pairs  of  such  classes  (up  to  all  the 
classes,  provided  there  is  no  odd  one  that  varies),  vary  to  opposite  geometric 
extremes  so  that  the  geometric  average  is  unity  ;  for  then  the  arithmetic  aver- 
age with  the  weighting  of  the  first  period  (or  the  harmonic  with  that  of  the 
second )  is  also  unity,  being  so  for  each  of  the  pairs  and  for  the  unchanged 
figures. 

It  holds  also  in  another  case,  which  may  be  shown  as  follows  for  three 
classes.     We  wish  to  prove  that  under  certain  conditions 


b^C2  ^  xa.^  +  ybj  +  ZC2 
bi    Ci       a-oj  4  ybi  +  zc^ ' 

a  given  condition  being  that  xa^  •  xa.^  =  yb^  ■  yb^  ^  zc^  •  zc^.     From  this  condi- 

^2/1  p  2  /*  C 

.  tion  we  obtain  a,  =  -~  and  6,  =:  ^r^  •     Upon  substitution  of  these  values 
'  x'^c/j  y^b^ 

in  the  equation  to  be  proved,  it  reduces  to 


{xa^Y-iyb^Y-  (sci)2      xai-yb^'zc^{xai  + yb^-{- zc^)  ' 
This  we  see  to  be  true  when  x(\  =  yby  =  zc^  ;  for  then 


1/ 


(xa^)s        (a-aj)3  3xai' 


which  is  evident.     But  then,  according  to  the  condition,  we  must  also  have 

ZO2  =  3/&2  =  2^2)  s^^d  both  these  conditions  together  mean  that  ^  =  r^  =  ^  1 

«i       ''1       ^ 
that  is,  that  the,  variations  must  all  be  alike. 

The  same  result  is  obtained  if  we  analyze  the  case  with  four  figures  or 
classes,  again  with  five,  and  so  on. 

We  know,  moreover,  the  correctness  of  this  result,  because  we  know  in 
general  that  when  variations  are  all  alike  any  average  of  them  with  any 
weighting  is  the  same  as  the  common  variation. 

§  9.  In  other  cases  trial  sliows  that  the  geometric  average  of  the  variations 
with  weighting  according  to  the  geometric  means  between  the  sizes  of  the 
classes  at  each  period  does  not,  in  ordinary  cases,  iinich  difTer  from  the  arith- 


VARIATIONS    AND    AVERAGES  521 

metic  average  of  them  with  weighting  according  to  the  sizes  of  the  classes  at 
the  first  period  or  (which  is  the  same  tiling)  tiie  harmonic  average  with 
weighting  according  to  tlie  sizes  at  the  second  period  (tliere  heing  a  common 
form  to  whicli  both  of  tliese  averages  so  weighted  reduce). 

More  specifically,  trial  seems  to  show  that  the  geometric  average  is  above 
the  other  common  form  when  the  preponderating  classes  have  variations  rising 
above  the  average,  and  below  the  other  common  form  when  the  prepondera- 
ting classes  have  variations  falling  below  the  average.^  Here  the  preponder- 
ance is  to  be  determined  by  comparing  the  sums  of  the  weights  (measured  as 
above  described)  of  tlie  former  with  those  of  the  latter  classes. 

The  divergence  of  the  geometric  average  from  the  common  form  seems  to 
be  greatest  either  when,  amidst  ordinary  variations,  the  variations  of  the 
classes  whose  combined  weights  are  about  one  fifth  of  the  whole  w'eighting 
are  nearly  the  same,  or  at  least  in  the  same  direction,  while  the  rest  are  in 
the  opposite  direction,  or  when  the  variations  of  the  smaller  classes  in  the 
same  direction  are  excessive.  In  all  other  cases  the  results  are  very  close 
together.  Especially  so  if  none  of  the  variations  be  great  or  unusual,  or  if 
great  and  extraordinary  variations  of  some  large  classes  are  met  by  great  op- 
posite variations  of  other  nearly  equally  large  classes,  or  of  many  small  ones 
in  combination  nearly  equalling  them,  or  if  the  great  variations  of  small 
classes  are  met  by  great  opposite  variations  of  nearly  equally  small  classes. 
In  general,  if  the  combined  weights  of  the  classes  rising  above  the  average 
and  the  combined  weights  of  the  classes  falling  below  the  average  are  nearly 
even,  these  conditions  tend  to  bring  the  results  together  ;  and  it  is  possible 
that  they  may  coincide. 

VII.  COMPARISON  OF  AVERAGES  OF  UNEQUAL  SETS. 

For  completeness  we  may  add  consideration  of  another  case.  So  far  Me 
have  all  along  supposed  that  the  numbers  of  figures  in  all  the  classes  separately 
and  together  are  the  same  at  both  periods  ;  for  only  in  this  case  can  all  the 
figures  at  the  second  period  be  regarded  as  variations  of  figures  at  the  first. 
But  it  may  happen  that  we  can  have  different  numbers  of  figures  in  the  whole 
sets,  or  even  a  different  number  of  classes  (since,  so  long  as  we  are  having  a 
different  number  of  figures,  it  may  not  matter  whether  the  new  ones  be  in  old 
classes  or  whether  they  be  altogether  new  ones  forming  new  classes)  ;  and  we 
may  still  have  reason  for  wishing  to  compare  the  averages  of  these  sets  of 
figures. 

This  case  can  be  represented  by  numbering  all  the  symbols  previously  left 
undistinguished.  The  numbers  of  classes  of  figures  at  the  two  periods  may 
be  represented  by  ii^  and  n.^  respectively  ;  the  numbers  of  times  the  figures 

ctj,  61,  occur  at  the  first  period,  by  Xj,  y^,  ,  and  the  numbers  of  times 

the  figures  «2,  ^21  occur  at  the  second,  by  x.,,  i/o,  ;   and  the  total 

numbers  of  the  individuals  at  the  first  period  by  u/,  so  that  «/  =  a-j  -\-  yi-\- 

to  rij  terms,  and  at  the  second  by  n./,  so  that  n./  ^=  x^  -\- y2  + to 

Wj  terms. 

2  If  the  geometric  average  shows  a  rise  from  1  to  2,  the  rise  of  a  figure  fi-om  1 
to  li  is  virtually  a  f\xll  compared  with  the  variations  of  all  the  figures ;  while  if 
the  geometric  average  shows  a  fall  from  1  to  J,  the  fall  of  a  figure  from  1  to  3  is 
virtually  a  rise  compared  with  the  variations  of  all  the  figures. 


522  APPENDIX    A 

§  1.  Arithmetic  averagin.cj.     The  average  at  the  first  period  is 

XM-,  +  w,&,  -I-    to  riq  terms  1     ,  ,       i     .  .         .  \ 

--   ,   ^'  \ 7 7 ,     or    — y   /!«!  +  i/i&i  + to  n^  terms ) ; 

J-,  +  2/i  + to  Wj  terms  n{ 

and  at  the  second  it  is  similar,  with  all  the  numbers  changed.     Therefore  the 
variation  of  the  averages  is 

-^  {x^a^  +  2/2^2  + to  ^^2  terms) 


A        1 

^       —^  ( '"if'i  4-  !/i''i  + to  »!  terms ) 

which  reduces  to 

A.,  _n/{x2a2  +  2/2^2+ to  rij  terms) 

A]  ~  /i2^(iiai  +  2/1^1  4~ to  ?ii  terms )  ' 

(Here,  incidentally,  we  see  that  the  variation  of  the  averages  is  not  the  vari- 
ation of  the  sums,  and  can  not  be  replaced  by  that,  since  the  variation  of  the 
sums  does  not  allow  here  for  the  change  in  the  numbers  of  the  figures. ) 

By  restoring  the  values  of  n/  and  n./,  and  rearranging,  the  expression  may 
be  stated  in  full,  as  follows  : 

A2 .tjCta  +  ■i/2'^2  "I" to  ?i2  terms      a"i  +  2/i  4- to  n^  terms 

Ai      ccjOi  -\-  y^bi  + to  Tij  terms     2^2  +  2/2  4" to  nj  terms  " 

§  2.  Harmonic  averaging.  This  being  worked  out  in  the  same  way,  the  vari- 
ation of  the  averages  reduces  to 

A  +v:  + to  7ii  terms  , 

A2 «i       Oj  3:2  -i-  2/2  -f- to  «.2  terms 

A,       ■»■•>,   i/a   ,  ^         X  ^1  4-  2/1  + to  w,  terms ' 

-'-{-,+ to  71,  terms        1    '  -^i   '  1 

^2*2 

§  3.   Geometric  averaging.     The  variation  of  the  averages  obviously  is 

A2 ^y  c^-?^  •  ^2^2 to  n^  terms 

1       "l/  tti^'  ■  i>x>^ to  «-!  terms 

§  4.  All  three  of  these  variations  of  averages  have  the  following  common 
properties. 

Without  any  variation  in  the  sizes  of  the  figures  between  the  first  and 
the  second  period,  a  variation  in  the  numbers  of  the  figures  in  the  classes — a 
change  of  weighting — may  jn'oduce  a  variation  in  the  average.  These  cases 
come  under  the  principles  in  I.  §§-8,  9,  10.  Thus,  without  any  variation  in 
the  sizes  of  the  figures,  an  increase  or  decrease  in  the  number  of  any  figure 
above  the  average  at  the  first  period  will  raise  or  lower  the  average  at  the 
second  period  ;  and  reversely  an  increase  or  decrease  in  the  number  of  any 
figure  below  the  average  at  the  first  period  will  lower  or  raise  the  average  at 
the  second  period  ;  while  any  change  whatever  in  the  number  of  a  figure 
equal  to  tiie  average  at  the  first  period  has  no  influence  upon  the  average  at 
the  second  period. 

Also,  there  being  some  variation  in  the  average  at  the  second  period  from 
the  first,  due  to  variations,  all  alike  or  otherwise,  in  the  sizes  of  the  figures 
(supposing  no  change  in  their  numbers),  then  a  superadded  change  in  the 
numbers  of  the  figures  may  accentuate,  lessen,  nullify,  or  outbalance  the 
variation  of  the  average  due  to  the  other  influences  alone. 


VARIATIONS    AND    AVKRACJES  023 

?  5.  Naturally,  if  the  imnihers  have  not  clianged  at  all  from  tlie  tirst  to  the 
second  period,  these  fornnihe  all  reduce  to  tlie  corresponding  foriiiulfe  witli  the 
same  numhers  at  both  periods. 

They  do  so  also  if  the  numbers  of  figures  in  the  classes  all  vary  in  the  same  pro- 
portion, and  the  number  of  the  classes  remains  the  same  (i.  e.  n.^  ^=  n^).  We 
have  then  the  state  of  things  described  in  I.  §11 — retention  of  the  same 
weighting  ;  and  so,  for  instance,  impossibility  of  a  variation  in  the  average 
without  any  variation  in  the  sizes  of  the  figures.  In  effect,  suppose  from  the 
first  to  the  second  period  every  number  increases  or  decreases  in  the  pro- 
portion /■ ;  then  .r.,  ^  rx^,  y.,  =  n/j,  and  so  on,  and  the  exjiression  for  the 
arithmetic  average  becomes 

A2 ri\(ii  -\-  ''2/1^2  4- to  n  terms      Xy-{-  2/1  + to  n  terms 

Ai        x^a^  +  1/16,  4- to  n  terms     r^Tj  +  rj/j  +  to  n  terms ' 

which  reduces  to 

A2  -I'lC^  "4"  Vi^'i  ~l~ to  «  terms 

Ai       a'lOi  4-  2/1*1  ~l~ to  TO  terms' 

that  is,  to  what  is  sometimes  the  proper  form  of  the  average  of  the  variations 
on  the  numbers  which  are  common  to  both  the  periods — in  this  case  the 
numbers  at  the  first  period.  Or  if  we  let  r^  be  the  reciprocal  of  r,  then 
Xj  =  r^X2 ,  2/1  =  '"'2/2 )  ^"f^  *^o  on,  and  the  expression  will  still  reduce  to  what 
is  in  the  same  cases  the  proper  form  of  the  average  of  the  variations  on  the 
numbers  which  are  common  to  both  the  periods — this  time,  the  numbers  at 
the  second  period,  wherefore  it  is  indifTerent  whether  we  use  the  numbers  of 
the  first  or  of  the  second  period.  And  similar  would  be  the  reductions  in  the 
expressions  for  the  variations  of  the  harmonic  and  the  geometric  averages, 
except  that  these  would  yield  forms  for  the  averages  of  the  variations  not 
proper  when  the  above  forms  of  the  arithmetic  averages  are  proper,  the 
weighting  of  the  variations  being  different. 

^  6.  In  all  other  cases  there  is  no  reduction  of  these  formulpe  to  any  for- 
mula' for  averages  of  variations.  It  is  impossible  to  find  any  formulae  for  any 
averages  of  variations  which  agree  with  any  of  the  above  formula  for  the 
variations  of  averages.  The  reason  is  simijle.  There  is  no  average  of  vari- 
ations in  these  cases,  because,  although  there  may  be  some  variations  con- 
tained in  the  two  sets  of  figures,  yet  there  are  other  figures  which,  appearing 
only  in  the  one  or  in  the  other  set,  do  not  represent  a  variation  of  anything. 
The  forms  here  reviewed  are  averages  of  the  figures  as  reported — in  the 
numbers  that  happen  to  be  reported.  In  some  matters  it  is  necessary  to  alter 
these  numbers  into  other  numbers,  in  various  ways,  always  according  to  some 
established  principle.  After  such  reduction,  the  new  numbers  being  used  in 
place  of  the  old,  the  formuhe  will  remain  the  same  in  form,  although,  being 
applied  to  different  numbers,  their  results  will  be  difierent.  But  it  is  some- 
times possible  to  introduce  into  the  formulae  themselves  the  principle  by 
which  the  reduction  of  the  numbers  is  made.  One  condition  for  this  is 
that  the  new  formulae  may  be  made  to  contain  something  which  necessarily 
confines  them  to  the  same  number  of  classes  at  each  period.  Then  the  for- 
mulae themselves  become  different  in  form.  And  now  the  new  formula'  may 
even  have  some  points  of  contact  with  the  formuhe  for  averages  of  variation. 


APPENDIX   B. 


ON   COMPENSATORY  VARIATIONS. 

The  supposition  is  that  we  have  two  series  of  three  terms,  a,  m,  h,  which 
are  such  that  a:^??i  =  6  at  one  period  and  at  another  a  and  b  are  opposite 
terms,  compensating  for  each  otlier,  in  one  of  the  three  progressions,  arith- 
metic, harmonic  or  geometric,  so  that  m  is  always  eitlier  the  arithmetic,  har- 
monic or  geometric  mean  between  a  and  b.  As  m  is  always  the  mean,  its 
presence  or  absence  in  the  calculations  is  indifferent ;  and  Ave  are  virtually 
dealing  with  two  figures,  a  and  b,  and,  treating  them  as  equally  important, 
are  using  even  weighting.  We  may  suppose  that  one  of  these  terms,  always 
a,  when  not  equal  to  m,  is  given,  and  that  this  term  is  always  larger  than  m 
(which  is  also  a  given  term).  Then  the  other,  always  smaller  than  m  at  the 
same  period  when  a  is  larger,  may  always  be  expressed  in  these  given  terms. 
Thus  when  the  terms  are  in  arithmetic  progression,   b  =  2m  —  a;  when  in 

,  .  .        ,  nm  ,        .  .  .        .        m^ 

liarnionic  progression,  o  = ;  when  in  geometric  itrogression,  0  = . 

2a  —  m  a 

We  may  suppose  either  (I)  that  the  terms  are  equal  at  the  firet  period  and 

a  and  b  become  divergent  at  the  second,  or  (II)  that  a  and  b  are  divergent  at 

the  first  period  and  by  converging  become  equal  at  the  second.     There  will 

then  be  occasion  for  some  explanatory  and  amplificative  remarks. 

I. 

§  1.  In  this  supposition  we  have  a^  =  m  =  b^  and  aj  >  m  >  ^2-  Here  aj 
and  ^2  have  varied /?'om  the  mean  to  opposite  extremes,  a.^  having  risen  and  b^ 
fallen. 

The  variation  of  «,  from  «,  is  always  -^  ^^  ^ .     The  variation  of  />.,  from 

%       m 

i    i     .1  -.1         •  ,  .     &o       2m  —  a, 

Oi  to  the  arithmetic  extreme  is  ,    =  :    to   the   harmonic   extreme, 

Wj  m 


.        jl  .      •  ^9  df  W*  -»T  1  .1 

_  ,  to  the  geometric,  ,   =  — =- ^^      .     Now  the  arithmetic 

2a2—m  "  ftj        ??!,         a., 

1    X  xi  •  X-        ,1         -1         .  .    1   /  cif,   ,    2m  —  a,  \ 

mean  between  the  variations  to  the  arithmetic  terms  IS   ,  (     "4-  "  1=1; 

2  \  m  m       J 

wlierefore  these  may  be  regarded  as   (simple)   arithmetic  variations  (whose 

arithmetic  mean    indicates  constancy).      Tlie  harmonic   mean   between   the 

524 


ON   COMPENSATORY    VARIATIONS  525 

variations  to  the  harmonic  terras  is ~ =  1 ;  wherefore  these 

1  />rt       iOj  —  m\ 

may  be  regarded  as  (simple)  harmonic  varicdions  (whose  harmonic  mean  indi- 
cates constancy).  Tlie  geometric  mean  between  the  variations  to  the  geomet- 
ric terms  is  -^ =  1  ;  wherefore  these  may  be  regarded  as  (simple)  geo- 
metric variations  (whose  geometric  mean  indicates  constancy). 

§  2.  The  percentage  (in  hundredths)  of  the  rise  of  a.j,  above  m,  reckoned  in 

m,  is  always  ^ .     The  percentage  of  the  fall  of  63  below  in,  likewise 

1                1        J-          •    ^1         -^1       .•          •  .•       •   "* — (2"i  —  02)       «2  — '" 
always  reckoned  in  m,  in  the  arithmetic  variation  is ^ —  --=  — ; 


^,       ,  .  ...  2a, — m       a,  —  m       .        , 

in    the   harmonic    variation, =  --^  ;    in    the    geometric, 

m  2a^  —  m 

m 

=  -^ .  Thus,  the  percentage  being  reckoned  in  the  mean  or  the  uni- 
form starting  point,  it  is  only  the  rise  and  fall  from  the  mean  to  the  arithmetic 
extremes,  or  the  arithmetic  variations,  that  are  in  equal  percentage. 

§  3.   But  if  we  reckon  the  percentage  in  the  terms  reached,  the  percentage 

of  the  rise  of  a.^,  in  o,,  is  always  — .     The  percentage  of  the  fall  of  b^, 

a^ 

,  .     ,     .      ,  .  ,         .  ...    m — (2m  —  a,)         a,  —  m      .      , 

alwavs  in  o„  in  the  arithmetic  variation  is ^ ~  =^  ~ ;  in  the 

2m  —  a,  2m  —  a, 


,              .           .     .                   2ao  —  m         a,  —  m     .      ,                    .  a., 

harmonic  variation,  =  — ;  in  the  geometric,  ; — -^ 

2a2  —  m  Oj 

~ .     Thus,  the  percentage  being  reckoned  in  the  opposite  extremes  reached,  it 

is  only  the  rise  and  fall  from  the  mean  to  the  harmonic  extremes,  or  the  harmonic 
variations,  that  are  in  eqwd  percentage. 

I  4.  If  again  we  reckon  the  percentage  always  in  the  same  direction,  that 
is,  (1)  from  the  lowest  terms,  or  (2)  from  the  highest  terms,  or  in  other  words, 
(1)  from  the  starting  point  at  the  mean  for  the  rising  term  and  from  the  ex- 
treme point  reached  for  the  falling  term,  or  (2)  from  the  extreme  point 
reached  for  the  rising  term  and  from  the  starting  point  at  the  mean  for  the 
falling  term,  we  have  the  following  percentages.     (1)  The  percentage  of  the 

rise  of  a^,  reckoned  in  m,  as  shown  in  ?  2,  is ;  and  the  percentages  of 

the  falls  of  ftji  reckoned  in  b^,  are  given  in  §  3.  We  see  that,  the  pe>-centage 
being  reckoned  in  the  louest  terms,  it  is  only  the  rise  and  fall  from  the  mean  to  the 
geometric  extremes,  or  the  geometric  variations,  that  are  in  equal  percentage.    (2)  The 

percentage  of  the  rise  of  Oj,  reckoned  in  a.,,  as  shown  in  iJ  3,  is  -^ ;  and 

the  percentages  of  the  falls  of  b.,,  reckoned  in  m,  are  given  in  §  2.  We  see 
that,  the  percentages  being  reckoned  from  the  highest  tenns,  it  is  only  the  rise  and  fall 


526  APPENDIX   B 

from  the  mean  to  the  geometric  extremes^,  or  the  c/eometric  v(triHtioiis,  tli(tt  are  in  equal 
percentage. 

11. 

H-  In  this  supposition  we  have  a^  >  m  >  /;i  and  «.,  :=  m  =  b.^  ;  and  a., 
and  bo  have  varied  to  the  mean  from  opposite  extremes,  «,  "ow  having  fallen 
and  b.,  risen. 

The  variation  of  a.,  from  a-,  is  always  -^  =      .     The  variation  of  />.,  from  6,, 

from  the  antlimetic  extreme,  is  r  =  h ;  ""o'"  "le  harmonic  extreme, 

Oi       Zm  —  Oj 

6,  m  2a, — in    »         _,,  ^  .        ,  b.^         m        a.^ 

^  = ^  — * ;  from  the  geometric  extreme,  y-  =  — -  =      .     i  hus 

6i  mai  «i  by        m^        m 

2aj  —  m  "i 

these  variations  are  the  reciprocals  of  the  preceding  ;  for  we  may  suppose  a^ 

•here  to  be  equal  to  o^  there,  and  b^  here  equal  to  b^  there. 

§  2.   Now  unity  is  the  harmonic  mean  between  the  variations  from  the 

arithmetic  extremes  ;  for  r— j: r-  =  1  ;  wherefore  the  variations  from 


2V™ 


+ 


2?ji  —  Oj^ 


the  arithmetic  extremes  to  their  mean  are  Imrmonic  variations  (whose  harmonic  mean 
indicates  constancy).     Unity  is  the  arithmetic  mean  between  the  variations 

from  the  harmonic  extremes  ;  for   -  ( 1 ^ )  =  1  !  wherefore  the  vari- 

2Voi  Oi       / 

ations  from  the  harmonic  extremes  to  tJieir  mean  are  arithmetic  rariations  (whose 
arithmetic  mean  indicates  constancy).  But  unity  is  the  geometric  mean  he- 
lm «-[  , 
tween  the  variations  from  the  geometric  extremes  ;  lor  -y  —  •  =  1  ;  where- 
fore the  variations  from  the  geometric  extremes  to  their  mean  are  geometric  variations 
(whose  geometric  mean  indicates  constancy). 

g  3.  In  effect,  if  we  reduce  the  terms  at  the  first  period,  a^  and  b^,  to 
equality  to  m,  and  observe  the  same  variations,  we  may  reduce  the  terms  at 
the  second  period,  a^  and  b^,  to  figures  no  longer  equal  to  m,  as  follows.     We 

reduce  a^  to  m  by  multiplying  a^  by      ;  therefore  we  must  reduce  a.,  in  the 

same  manner  ;  but  a,  •  = —  .     Thus  the  variation  of  «,  to  m  is  the  same 
"    a^       «! 

m^  .  m 

variation  as  that  of  m  to  —  .     We  reduce  6j  to  m  by  multiplying  6i  by       ; 

and  we  must  reduce  b^  in  the  same  manner.     Now  when  the  terms  were  in 

m 
arithmetic  progression,  we  had  &j  =  2»}i  —  a-^  ;  therefore  we  have  b^-      ~m  ■ 

"\ 

=  - .     Thus  the  variation  from  5.,,  when  it  is  at  the  arithmetic 

2m  —  a      2m — a^ 

m^ 
extreme,  or  from  2m —  a,  to  m  is  the  same  variation  as  that  of  m  to  „       —  . 
'  '  2m  — «! 

Similarly  when  the  terms  were  in  harmonic  progression,  the  variation  from  tlie 

harmonic  extreme,  b,,  to  the  mean,  or  from ^—  to  m,  is  the  same  as  that 

2a — m 

m(2a  —  m)         .      ,      ,         , 
from  m  to  — ^ .     And  wlien  tlic  terms  were  in  geometric  i)rogression. 


ON    COMPENSATORY   VARIATIONS  527 


the  variation  from  tiie  geometric  extreme,  6,,  to  tiie  mean,  or  from  —  to  m, 
is  tlie  same  variation  as  that  of  ni  to  a^ .     Now  —  and  ^  are  tlie  har- 

monic  terms  aroinul  m  ;  for  — — = r  =  m  .     Therefore    variations 


1  I  a^       2m,  —  Oj  \ 

2  Vm^"*        rn^      ) 


from  the  arithmetic  extremes  to  their  mean  are  the  same  as  variatians  from  the  mean  to 
h^triiwnic  extremes.  Consequently  it  is  the  percentages  of  these  variations, 
reckoned  in  the  extremes,  the  positions  originally  at  the  first  period,  but  at 

the  second  period  in  the  reductions,  tliat  are  equal.     And  —  and ^ 

are  arithmetic  terms  around  m  /  for  half  the  sum  of  these  is  m.  Therefore 
variations  from  the  harmonic  extremes  to  their  mean  are  the  same  as  variations  from 
the  mean  to  arithmetic  extremes.  Consequently  it  is  the  percentage  of  these  vari- 
tions  reckoned  in  the  mean,  the  position  originally  at  the  second  period,  but 

at  the  first  period  in  the  reductions,  that  are  equal.     But   —  and  a^  are  geo- 

% 

metric  terms  around  m  as  their  mean  ;  for   v  —  •  a-,  =  m  .     Therefore  varia- 

tionsfrom  the  geometric  extremes  to  their  mean  are  the  same  as  variations  from  the 
mean  to  geometric  extremes.  Consec^uently  it  is  the  percentages  of  variations 
reckoned  either  from  the  top  or  fi-om  the  bottom  that  are  equal. 

§  4.  Thus  if  two  quantities,  equal  at  first,  vary  to  the  opposite  arithmetic 
extremes,  and  then  vary  back  to  the  same  figure  as  at  the  start,  their  first  vari- 
ations, diverging,  are  arithmetic  variations,  and  their  later  variations,  con- 
verging, are  harmonic  variations.  If  two  quantities,  equal  at  first,  vary  to 
the  opposite  harmonic  extremes,  and  then  vary  back  to  the  same  figure  as  at 
the  start,  their  first  variations,  diverging,  are  harmonic  variations,  and  their 
later  variations,  converging,  are  arithmetic  variations.  But  if  two  quanti- 
ties, equal  at  first,  vary  to  the  opposite  geometric  extremes,  and  then  vary 
back  to  the  same  figure  as  at  the  start,  both  these  diverging  and  converging 
variations  are  geometric  variations. 

HI. 

The  following  explanations  may  be  offered. 

§  1.  In  I.  we  have  been  considering  variations  of  states  which  may  be 
represented  thus, 

02  >  TO  >  &2 


;  m  =  6j 


But  we  have  been  averaging  the  variations,  with  even  weighting.  Hence,  in 
accordance  with  Appendix  A,  I.  §  7  (II.  §6,  III.  §6,  and  V.  §2),  we  have 
been  performing  the  same  operation  as  if  we  averaged  the  figures  at  each 
period  separately,  with  even  weighting  in  each  case,  and  then  compared  the 
results. 

§  2.  In   II.  we  have  been  considering  variations  of  states  which  may  be 
represented  thus, 


528  APPENDIX    B 

flj  ">  m  ^  bi       in         ' 

in  each  of  which  averages  even  weighting  is  employed.  But  we  have  been 
averaging  the  variations  with  even  weighting,  and  noAV  the  operations  are  not 
the  same  in  two  out  of  the  three  cases. 

§  3.  Thus  if  the  terms  change  from  the  arithmetic  extremes,  applying  the 
arithmetic  average  with  even  weighting  to  the  figures  at  each  of  the  periods 
separate]}',  we  have 

A2  i{m  -\-  m)     '^" m 

But  we  do  not  obtain  this  result  by  arithmetically  averaging  the  variations 
with  even  weighting  ;  for 


1/m  ^ri       \        

2  V«i       2m  — ni/       a^{2r, 


but  only  by  harmonically  averaging  them,  if  we  are  to  use  even  weighting, 
as  shown  in  II.  ?  3.  This  is  because  the  comparison  of  the  arithmetic  aver- 
ages with  even  weighting,  when  the  figures  are  unequal  at  the  first  period,  is 
the  same  as  the  arithmetic  average  of  the  variations,  not  with  even  weighting, 
but  with  weighting  according  to  the  sizes  of  the  terms  at  the  first  period, 
according  to  Appendix  A,  II.  ?  7  ; '  and,  according  to  Appendix  A,  IV.  'i  2, 
this  is  the  same  as  the  harmonic  average  of  the  variations  with  even  weight- 
ing. 

§  4.  If  the  terms  change  from  the  harmonic  extremes,  applying  the  har- 
monic average,  we  have 


-('  +  -) 

2  V  1^1       'nt'  ) 


A, 

Ai \ 

2ay 


2\a 


-  + 


And  again  we  do  not  obtain  this  result  by  harmonically  averaging  the  varia- 
tions ;  for 

.J- 1  m(2aj  —  m)  _ 


1   (a^ 

2 


\  jft       2ai  —  m  ) 


but  only  by  arithmetically  averaging  them,  if  we  are  to  use  even  weighting, 
as  shown  in  II.  §  3.  This  is  because  the  comparison  of  the  harmonic  aver- 
ages with  even  weighting,  when  the  figures  are  unequal  at  the  first  period,  is 
the  same  as  the  harmonic  average  of  the  variations,  not  with  even  weighting, 
but  with  weighting  inversely  according  to  the  sizes  of  the  terms  at  the  first 

1  Here,  with  even  weighting,  with  a  limited  between  m  and  2m,  the  result  is 
larger  than  unity,  indicating  a  rise.  This  is  because  the  average  of  the  variations 
can  be  brought  down  to  unity  only  by  giving  greater  weight  to  the  fall  of  rtj. 


ON   COMPENSATORY    VARIATIONS  529 

period,  according  to  Appendix  A,  III.  §  7  ;  '■'  and,  according  to  Appendix  A, 
IV.  §  5,  this  is  the  same  as  tlie  arithmetic  average  of  tlie  variations  with  even 
weighting. 

§  5.  But  if  the  terms  change  to  the  geometric  extremes,  we  have 

A2 yin  -m  m 

^         «i 
and  we  obtain  the  same  result  by  geometrically  averaging  the  variations  with 
even  weighting,  as  shown  in  II.  ^  3.     This  is  because  the  comparison  of  the 
geometric  averages  is  the  same  as  the  geometric  average  of  the  variations  with 
the  same  weighting,  according  to  Appendix  A,  V.  §  5. 

IV. 

The  consequences  indicated  in  II.  ^  4  may  be  amplified  as  follows. 
§  1.  Suppose  we  have  these  three  series  at  three  difi'erent  periods  : 


1st  period — 

ai=m=^bi; 

2d       " 

«2  >  ""i  >  h ; 

3d       " 

a,  =:  m  =  b,  ; 

in  which  a.^  =  a^  and  b-^  =  b^.     Here  we  have  two  variations  of  the  series, 
which,  with  substitution  of  a^  for  03  and  of  b^  for  63,  are 

.     •  a2^>m^  bn 

1st  variation —     — 


2d 


ttj  =  m  =  bi 
Oj  =  m  =  6^ 


Oj  ^>  m  >  62 

But  the  two  variations  together  produce  a  return  to  constancy  ;  for 
03  =  ni 


.b. 


For  what  follows,  however,  it  is  not  necessary  that  a^  and  b.^  be  on  opposite 
sides  of  m.  They  may  be  any  quantities,  with  any  variation  of  their  mean 
(now  to  be  represented  by  m^)  from  m^.  All  that  is  needed  is  that  they  return 
at  the  third  period  to  what  they  were  at  the  first ;  whereupon  also  their  mean, 
OT3,  will  return  to  m^. 

I  2.  If  we  average  these  series  separately  and  then  compare  the  averages, 
we  get  inverted  results,  which  are  reciprocals  of  each  other.  Then  the  com- 
parison of  the  third  period  with  the  first  by  means  of  the  two  intermediate 
comparisons — which  comparison  consists  in  multiplying  together  the  results 
of  those  comparisons — turns  out  correctly  ;  for  these  reciprocal  results  multi- 
plied by  each  other  give  unity. 

§  3.   We  have  two  sets  of  variations,  which  are 

2  Here,  with  even  weighting,  with  a  larger  than  m,  the  result  is  smaller  than 
unity,  indicating  a  fall.     This  is  because  the  average  of  the  variations  can  be 
brought  up  to  unity  only  by  giving  a  greater  weight  to  the  rise  of  b^^. 
34 


530  APPENDIX    B 

1st  set  of  variations- 
2d    " 


«i      ^1 

"l      ^  . 

«,      0, 


and  tlie  wliole  variations  from  the  first  to  the  third  ijeriod  are    -^  =  —  :=1  and 

-  ^  -'==  1  ;  wherefore  the  average  of  these  whole  variations  is  always  unity, 

indicating  constancy.    But  if  we  average  each  set  of  variations,  and  seek  thence 
the  average  of  the  whole  variations,  we  obtain  the  following  results. 

^  -4.  The  arithmetic  averages  of  the  variations,  with  even  weighting,  are, 

of  the  first  set,  tt  I  ^  +  ,^  )  ,  ^"(^  of  the  second,  v.  (  ^  +  y^  I  •     The  multipli- 
2  \oi       6,/  2  Voj       62/ 

cation  of  these  gives  o  +  ^  (  ~x    1 T  )  '  ^^'"i*^^i  oy  elimination  of  a^  and  Oj, 

which  are  equal,  reduces  ^o^-\--j(  —  -\-~).     This  does  not  reduce  to  unity 

unlesss  «2  =  ^2'     ^^  ^^^  other  cases  its  result  is  above  unity,^  indicating  a  rise, 
instead  of  constancy. 

§  5.  The  harmonic  averages  of  the  variations,  with  even  weighting,  are,  of 

the  first   set,  ~ — ; r^-,  and  of  the  second  - — ; i^^^-     The  product 

2V«2       *J  2\a,^bJ 

of  these  is  :, — -  ^^  ,  which  does  not  reduce  to  unity  except,  as  he- 


1    i/«2^/,Y 

2   ^    4    V  ^2  «2  / 


2 

fore,  in  the  simple  case  when  O2  ^=  b.^.     In  all  other  cases  its  result  is  below 
unity,  indicating  a  fall,  instead  of  constancy. 

§  G.  The  geometric  averages  of  tlie  variations,  with  even  weighting,  are,  of 

the  first, -v/  ^  •  ,'^,  and  of  the  second,  -\/    —  ,'.      The    product   of    these   is 

*  'h    "i  "2    "2 

unity,  indicating  constancy.     Therefore  the  continuous  use  of  the  geometric  mean 
through  both  the  variations  gives  the  right  reaultfor  the  tivo  variations  together. 

§  7.  But  again,  if  we  used  the  arithmetic  average  of  the  variations  for  the 
first  set  and  the  harmonic  for  the  second,  or  the  harmonic  for  the  first  and  the 
arithmetic  for  the  second,  all  with  even  weighting,  the  product  of  the  results 
thereby  obtained  give  unity,  indicating  constancy.  Therefore  the  alternate 
tise  uf  the  arithmetic  and  of  the  harmonic  means  of  the  variation  gives  the  right  result 
for  the  two  variations  together. 

This  proposition  is  plain  if  we  remember  that  the  harmonic  average  of 
given  figures  is  the  reciprocal  of  the  arithmetic  average  of  their  reciprocals, 
and  reversely  ;  and  that  when  variations  take  place  from  the  mean  to  the 

^  To  demonstrate  this  it  is  necessary  to  show  that  -  +  -  >  2.     Now  t  +  -  = 

t>     a  b     a 

a^  +  b^      ,    ...  .  ,         ^,       a2  +  62     2ab  „       ,  (a- 6)  2 

r — .     Let  tins  equal  x.    Then r — ■ ;-  =  a;  —  2  ;  whence  ^ 7—    =  .r  —  2. 

ab  ab  nb  ab 

(a—  6)2 
But r —  is  always  positive,  a  and  b  being  positive.     Therefore  x  — 2  is  posi- 
tive, and  X  is  greater  than  2. 


ON    COMPENSATORY    VARIATIONS  531 

arithmetic  or  to  the  liarmonic  extremes,  the  return  variations  from  these  ex- 
tremes to  the  mean  arc  tlie  reciprocals  of  the  first  variations. 


I  1.   Lastly  we  may  analyze  the  variations  in  the  following  series  : 
1st  period —     Oj  >  m  >  b^  ; 
2d  "  ",'-6,)<'«<&:,(-=^',); 

in  which  m  represents  any  one  of  the  three  means.  Here  the  figures  are  sup- 
posed simply  to  change  places,  the  one  falling  to  the  other  and  the  other  ris- 
ing to  it.    The  variations  are     '  and  . ' ,  which  are  reciprocals  of  each  other. 

«i  ^1 

§  2.  It  is  plain  that  when  Oj  is  becoming  equal  to  ft, ,  it  must  pass  through 
the  mean  between  them,  that  is,  it  must  fall  first  to  the  mean  and  then  from 
the  mean  ;  and  when  ft,  is  becoming  equal  to  a^ ,  it  must  pass  through  the 
mean  between  them,  that  is,  it  must  rise  first  to  the  mean  and  then  from  the 
mean. 

§  3.  Therefore  if  the  mean  be  the  arithmetic,  the  approaches  of  «,  and  h^ 
to  this  mean  are  harmonic  variations  (above,  II.  §  2),  and  their  departures 
from  it  are  arithmetic  variations. 

If  the  mean  be  the  harmonic,  the  approaches  of  o,  and  b^  to  this  mean  are 
arithmetic  variations,  and  their  departure  from  it  are  harmonic  variations. 

But  if  the  mean  be  the  geometric,  both  the  approaches  to  it  and  the  de- 
partures from  it  are  geometric  variations. 

.?  4.  Therefore  also  the  whole  variations  of  Oj  to  5j ,  and  of  b^  to  '(, ,  are 
geometric  variations.  This  is  plain  also  from  the  fact  that  the  variations  are 
reciprocals  of  each  other.  It  may  be  shown,  further,  in  the  following  way. 
The  percentage  of  the  rise  of  ft,  to  a,  may  be  represented  as  a  rise  by  p  per 
cent,  (reckoned  in  ftj).     Then  the  percentage  of  the  fall  of  a^  toft,  (I'eckoned 

in  n,),  being  a  fall  bv  p  from  1  +  >>,  is  a  fall  bv  ^   - —    per  cent.     The  rise  of 

bi  is  the  same  variation  as  the  rise  of  1  to  1  +  y) ,  and  the  fall  of  «,  is  the  same 

'variation  as  the  fall  of  1  to  1  —  ,     —  —  , ;  that  is,  the  rise  and  the  fall 

1  -j-  p       I  +  p 

are  from  1  to  geometric  extremes,  and  are  geometric  variations. 

§  5.  Moreover  all  geometric  variations  in  opposite  directions  are  expres- 
sible as  rises  and  falls  to  reciprocals,  that  is,  as  rises  from  1  to  1  +  /)  and  as  falls 

from  1  to  -  J—  .     Therefore  all  geometric  rarintions  in  opposite  (lirectionH  ore 

the  same  as  variations  from  some  one  figure  to  another  and  from  the  latter  to  the  for- 
mer. 

§  6.  Still,  in  these  geometric  variations,  «,  in  becoming  6j  and  6,  in  be- 
coming flj  must  pass  through  the  arithmetic  and  the  harmonic  means.  There- 
fore all  geometric  varialiona  may  be  vieved  «,s  composed  of  tivo  I'ariatioiis,  the  one 
harmonic  and  the  other  arithmetic,  in  either  order  as  we  please. 

?  7.  If  a  and  b  vary  at  the  same  uniform  speed  so  that  a^  becomes  6j  and 
ftj  a,  in  the  same  time,  they  pass  through  the  arithmetic  mean  at  the  same 
moment,  which  is  the  halfway  point  in  the  time  for  tlie  whole  alternation. 


o32  ATM'KNDIX     B 

If  «  and  b  vary  at  tlie  corresponding  uniform  rates  so  that  a^  becomes  b^  and 
6j  Oi  in  the  same  time,  they  pass  through  the  geometric  average  at  the  same 
moment,  which  also  is  the  timal  lialfway  point.  There  is  a  uniform  varia- 
tion M'hich  will  carry  them  through  the  harmonic  mean  at  the  same  moment 
at  the  timal  lialfway  point. 

I  8.  If  the  variations  be  the  ones  at  the  uniform  rates,  with  a  and  b  pass- 
ing through  the  geometric  mean  at  the  timal  halfway  point,  the  rising  figure, 
b,  crosses  the  harmonic  mean  first  and  the  arithmetic  mean  last,  and  the  fall- 
ing figure,  a,  crosses  the  arithmetic  mean  first  and  the  harmonic  mean  last. 
At  every  moment  a  and  6  are  at  geometric  extremes  around  their  halfway 
point.  Therefore  (because  of  Appendix  C,  YI.  ^  2)  at  the  moment  when  b 
crosses  the  harmonic  average,  a  crosses  the  arithmetic,  and  at  the  moment 
when  b  crosses  the  arithmetic,  a  crosses  the  harmonic. 


APPENDIX   C. 

REVIEW  AND  ANALYSIS  OF  THE  METHODS  EMPLOYING 
ARITHMETIC  AVERAGING  FOR  MEASURING  VARI- 
ATIONS IN  THE   EXCHANGE- VALUE 
OF  MONEY. 

Arithmetic  averaging  is  the  common  feature  in  so  many  widely  differing 
methods  of  calculating  variations  in  the  exchange-value  of  money  from  the 
variations  of  prices,  that  it  may  be  well  to  review  these  methods  by  expound- 
ing them  all  in  the  same  system  of  notation  ;  for  many  have  Ijeen  made  with- 
out any  formidation  at  all,  and  the  others  have  been  formulated  wit li  so  many 
different  symbols  that  comparison  is  difficult.  Also  some  of  the  methods  are 
so  complex  that  analysis  is  necessary  to  disclose  their  real  nature. 

The  notation  here  employed  is  that  employed  in  the  body  of  this  work.  Prices 
of  the  classes  [A],  [B],  etc.,  are  represented  at  the  first  and  second  periods  re- 
spectively by  numbered  Greek  letters — «i,  «2J  ;^i)  /^2i •  These,  unless  other- 
wise specified,  are  prices  of  the  ordinary  mass-units  used  in  commerce.  When 
prices  are  taken  at  the  first  period  all  as  1.00  and  the  prices  at  the  second 
period  are  reduced  accordingly,  the  Greek  letters  are  accented,  thus,  «/  = 

/?/= (=^1.00),  and  a^\  ji/,  (these  figures  by  their  excess  above, 

or  deficiency  below,  1.00  directly  indicating  the  variation  of  the  prices  ;   for 

a/  =z   -^  ^^'  =  -?,  and  so  on).     The  numbers  of  tliese  mass-units  (or  masses 

whose  prices  are  used)  in  the  classes  respectively,  as  they  are  supposed  to  oc- 
cur in  trade,  or  amounts  all  reduced  in  the  same  proportion,  are  represented 

by  X,  y,  ,  or  .1-^,  y' ,  ,  (which  are  the  same  symbols  as  in  Appendix 

A,  while  fl,  /?,  here  take  the  place  of  o,  6,  as  there  used).     These 

letters,  x,  y,  ,  we  must  notice,  do  not  represent  the  weights  of  the  classes 

(except  in  the  averagings,  generally  eliminated,  at  the  different  periods  sep- 
arately ;  they  never  represent  the  weights  of  the  price  variations).  The 
weights  of  the  price  variations  are  represented  by  a,  b,  for  the  varia- 
tions of  the  prices  of  the  classes  [A],  [B],  respectively.     Tliese  letters, 

a,  b,  ,  represent,  siipposedly,  the  numbers  of  ideal  individuals  in  the 

classes,  or  amounts  reduced  in  the  same  projwrtion, — that  is,  the  relative  sizes 
of  the  classes  measured  by  their  exchange-value  (or  money-value)  impor- 
tance.    In  other  words  a  ^  ■'«,  b  =  y/?,  and  so  on,  ora:b: ::  ■'« '.  f//? 

533 


0,34  AI'I'EXDIX    (• 

without  specifying  at  which  period,  or  how,  the  prices  are  used.  The  num- 
ber of  classes  used  in  the  calculation  is  represented  by  n.  The  number  of 
actual  individual  prices  (those  of  the  mass-units  employed)  in  all  the  classes 

together,  by  »';  so  that  ?i'  =  x  -!-//+   to  ?i  terms.     The  number  of  ideal 

individual  prices  in  all  the  classes  together,  by  n/^;  so  that  i/^  ^  a  +  b  -|- 

to  >i  terms.     The  terms  x,  //,  ,  a,  b,  ,  n,  r/  and  //',  when  the 

same  for  both  the  periods  compared,  need  not  be  numbered.  But  when  they 
are  employed  distinctively  for  the  conditions  at  one  period  alone,  they  are 
also  to  be  distinguished  by  subscript  numbers. 

I.    DUTOT'S   METHOD. 

Dutot,  1738  ;  see  B.  2,  especially  pp.  370-373. 

The  prices  of  articles  are  taken  at  their  market  quotations  on  the  mass- 
units  that  happen  to  be  employed  by  merchants,  however  large  or  small,  and 
are  added  together,  the  sums  for  the  two  periods  being  compared.  It  may  be 
exi)ressed  thus, 

P2  «2+/?2  + 


P,  «1  +  /?!  + 


Here  only  sums  are  used.  But  we  know  that  this  gives  the  same  result  as 
if  the  arithmetic  average  were  drawn  for  each  period  (see  Appendix  A,  II. 
§  1).     We  also  know  that  it  contains  hidden  weighting  for  the  price  variations, 

viz.  according  to  «,,  ;^j,  ,  with  arithmetic  averaging,  (see  Appendix  A, 

II.  ?7).  Therefore  this  method  contains  what  we  have  called  haphazard 
weighting  (in  Chapt.  IV.  Sect.  II.  §2). 

This  method  has  since  been  employed  by  Levasseur,  B.  18,  pp.  179,  180, 
and  by  A.  Walker,  B.  27,  even  in  series  of  many  years  ;  by  a  writer  in  the 
London  Bankers'  Macjnzine,  B.  37  ;  by  Krai,  B.  98  ;  and  by  Fraser  and  Sergei, 
B.  120. 

( As  the  mass-quantities — the  mass-units  hit  upon — are  constant,  this  method 
is  really  a  variety  of  Scrope's  method,  to  be  treated  of  later.  But  as  it  was 
invented  and  has  been  used  without  any  reference  to  the  mass-quantities,  it 
deserves  to  be  classed  by  itself. ) 

II.    CAllLI'S  METHOD  AND   ITS  VARIETIES. 

^  1.  Carli,  1766  ;  see  B.  4,  especially  pp.  350-351. 

All  the  prices  being  taken  at  the  first  period  as  equal  to  1.00,  the  prices  at 
the  second  period  are  taken  on  the  same  masses,  and  the  percentages  of  their 
variations  noted  ;  then  the  average  is  drawn  between  these.  The  procedure 
in  full  is  described  in  the  following  foriiuila,  upon  the  pri(!es  of  any  mass- 
units.     The  average  percentage  is  represented  by  %. 

in  wliicli  the  answer  indicates  a  rise  if  positive,  a  fall  if  negative,  and  con- 
stancy if  zero. 

The  same  result  is  yielded  by  the  following  formula, 

P,  _1   fa         ,3,    ,  \ 

Pi         n  V",^/^,    '    )' 


REVIEW    OF    ^rETIIODS  535 

in  whicli  the  answer  indicates  constancy  if  unity,  a  rise  if  above  unity,  a  fall 
if  below  unity  ;  and  indicates  the  percentage  of  the  rise  in  the  second  decimal, 
and  the  percentage  of  the  fall  in  tlie  second  decimal  of  the  remainder  when 
the  answer  is  snbstracted  from  unity.  If  the  prices  at  the  first  period  are  all 
taken  at  1.00,  and  those  at  the  second  period  are  reduced  accordingly,  the 
same  result  is  given,  in  the  same  way,  by  the  following, 

p;=»«+'V+ ). 

which  is  practically  the  form  erajjloyed  by  Carli.  Or  if  the  prices  at  the  first 
period  are  all  taken  at  100,  by  this, 

^=J  (ioo«/  +  ioo/v+ ), 

in  which  the  answer  indicates  constancy  if  100,  and  the  percentage  of  a  rise 
or  fall  according  to  the  units  it  is  above  or  below  100. 

This  last  form  of  the  method — the  one  which  has  usually  been  employed 
— was  introduced  by  Evelyn,  1798  ;  see  B.  5.  This  variation  of  Carli' s 
method  is  only  formal,  and,  when  it  is  employed,  the  method  still  deserves  to 
be  called  Carli' s  method.  In  it,  it  is  plain  that  when  we  are  comparing 
only  a  second  with  a  first  period,  there  is  even  weighting  of  the  price  varia- 
tions. 

^  2.  Evelyn,  however,  compared  several  periods  all  with  the  same  first 
period  (some  of  the  second  periods  being  later  and  some  earlier  than  the  com- 
mon first  or  basic  period )  ;  and  this  also  has  become  the  usual  practice.  Now 
in  this  case — confining  our  attention  to  the  simplest  form  of  the  formulae,  that 
in  which  prices  are  reduced  to  1.00  at  the  basic  period, — after  comparing  a 
second  period  with  the  first  and  getting  this  result, 

f; ='.«+''■''+ >• 

and  comparing  a  third  period  with  the  same  first,  with  this  result, 

?;  =  ^<+'V  + )> 

if  we  compare  these  with  each  other,  our  comparison  of  these  is  now  as  fol- 
lows, 

p^^a/+  l3/-\. 

Pi      «/+/?2'+ ' 

or 

3     ,     .    3     1      


But  here  we  know  that  the  weighting  of  the  price  variations  is  no  longer  even, 

— it  is  accidental,  being  according  to  a./,  j-i/ ,  or  -"^ ,    1^ ,  ,  that  is 

according  to  the  price  variations  between  the  base  and  the  earlier  period. 


536  APPENDIX    C 

Thus  this  method — mostly  without  the  knowledge  of  those  who  employ  it — 
switches  off  into  something  quite  different. 

This  method  of  Carli  was  so  employed  by  Porter,  B.  11  ;  and,  with  Eve- 
lyn's formal  variation,  it  has  been  employed  by  J.  P.  Smith,  B.  7. ;  the  Econo- 
mist, on  the  lists  of  prices  begun  by  Newmarch,  B.  19  and  20  ;  Laspeyres,  B. 
25  and  26  ;  Bourne,  at  first,  B.  46  ;  Bnrchard,  B.  53  ;  Hansard,  B.  67  ; 
Sauerbeck,  B.  79-90  ;  Rogers,  B.  92,  pp.  787,  789,  790,  791  ;  Falkner,  partly, 
B.  111-113;  Wiebe,  partly,  B.  124,  pp.  369-382;  Whitelaw,  B.  130,  pp.  32- 
33  ;  Wetmore  in  B.  119  ;  Barker,  B.  128  ;  Powers,  partly,  B.  131,  pp.  27,  28, 
29,  etc.  Geyer  employed  it  with  another  formal  variation,  reducing  the  first 
prices  to  about  2.00,  B.  23,  pp.  321-322. 

ij  3.  Evelyn  introduced  furthermore  a  real  variation.  He  counted  always 
wheat  by  itself,  butcher's  meat  by  itself,  and  day  labor  by  itself,  all  as  equally 
important,  and  also  as  equally  important  with  each  of  these  a  figure  made  up 
of  the  prices  of  twelve  other  agricultural  products,  each  of  which  had  been 
counted  as  of  equal  importance  in  making  up  this  figure.  This  is  irregular 
weighting  by  classification  ( in  which  classes  count  more  the  fewer  there  are  in 
the  divisions).  It  is  not  worth  while  to  give  a  formula  for  this.  It  would  be 
exceedingly  complicated  if  we  attempted  to  find  the  -weighting  employed  in 
the  comparison  of  two  later  periods  Avith  each  others.  Whenever,  with  the 
rest  of  Carll's  method,  instead  of  absolutely  even  weighting  in  the  comparison 
of  the  other  periods  with  the  basic  one,  there  is  employed  such  irregular 
weighting  by  classification,  this  may  be  called  Evelyn' s  method,  as  it  employs  a 
feature  really  distinctive  in  Evelyn's  procedure. 

This  method  was  sometimes  employed  by  Young  (who  in  doing  so  aban- 
doned his  own  method),  B.  6,  pp.  84,  118  ;  also  by  Soetbeer,  B.  19  ;  by  Krai, 
B.  98  ;  and  by  Wiebe,  partly,  B.  124,  pp.  383-386. 

III.    YOUNG'S  METHOD   AND   ITS  VARIETIES. 

?  I.   Arthur  Young,  1812  ;  see  B.  6,  p.  72. 

Prices  at  the  first  or  basic  period  are  taken  at  100  and  at  the  other  periods 
reduced  accordingly.  Percentages  of  the  variations  are  used,  and  these  are 
multiplied  by  the  weights  (according  to  the  relative  total  exchange-values  of 
the  classes,  in  general,  at  no  particular  period),  and  the  sum  of  the  products 
divided  by  the  sum  of  the  weights.     Tluis 

100  /     «,  —  a.    ,    .   ,'3,  —  ,3,    ,  \ 

%=,,{&-  '  4-  b    -  :,--^  +    to  n  terms), 

in  which  the  answer  has  the  same  meaning  as  in  C'arli's. 

The  same  result,  with  our  usual  interpretation  of  the  answer,  is  yielded 
by  the  following, 

Pj        1    /    «..    ,   ,   ^ 
P, 


Pj  1     /      «•.     ,     -u   /''•.     ,  ^ 

tT  =^  ~7/  I  S*    "  +  b   j"  -H to  n  terms 

Pi       n''  \    «i  /i,  J 


or,  if  we  start  with  prices  at  1.00,  by 

P         1 

TT  =^^  -T/  ( a  "./  +  b  ,1/  -I- to  ?!  terms) . 

J  I        ir  -  - 

Either  of  tliese  may  be  regarded  as  the  fornuda  for  Young's  method. 


REVIEAV    OP"    METHODS  537 

This  method  was  approved  and  recoiiimeiuled  by  Lowe,  B.  8  ;  and  lias  since 
been  reinvented,  with  slight  variations  in  the  procednre,  hy  Ellis,  B.  36,  and 
Wasserab,  B.  105  ;  by  Sauerbeck  at  times  (using  the  weights  of  1889-1891), 
B.  82,  p.  218,  and  later  ;  by  Falkner  at  times  (using  the  weights  of  periods 
at  the  end  of  the  series),  B.  112,  p.  63  ;  by  Fonda,  B.  127,  pp.  160-161  ;  and 
a  correct  formula  has  been  made  for  it  by  Westergaard,  B.  110,  p.  220,  in  this 
cumbrous  form, 

P2_  a Oj  J b /^j  _|_ 

Pi^a  +  b  +  • ■  «,  "^a-l-bH- *  /^i      ' 

which  easily  reduces  to  the  simpler  form  above  given. 

^  2.  Young  himself  happened  not  to  do  so,  but  the  rest  of  these  writers 
have  employed,  or  recommended,  this  method  in  a  series  of  periods,  all  the 
later  being  compared  directly  with  the  first.  Then  a  comparison  between  any 
of  the  later  periods  would  be  as  follows, 

Ps  __  a  a.,^  +  b/V+  •■■•  • 
P^     a«/  +  b/V4- '•' 

or,  with  the  prices  of  any  ma.ss-units, 

p       a^^'  +  bj+-- 


^'     a"^  +  b^^+ ' 

«i         Pi 

in  which,  as  we  know,  the  weights  of  the  price  variations  are  no  longer  a,  b, 

,  but  a«./,  b;V.  )  ora    ■,  b  '  %  (see  Appendix  A,  II.  ?  8). 

o,         /:(, 

For  this  difference  between  the  comparisons  of  the  later  periods  with  one  an- 
other and  the  comparisons  of  each  of  the  later  periods  with  the  first,  Wester- 
gaard has  found  fault  with  this  method  (see  here  Ch.  V.  Sect.  VI.  ?  7). 

I  3.  Variations  upon  this  method  have  been  made  by  GifTen  and  by  Pal- 
grave.  Giffen's  variation  is  mostly  formal,  though  it  is  real  as  regards  the 
choice  of  weighting.     The  formula  for  it,  for  the  second  period,  is 

%  =  1^  |a7(«,/  —1)4-  b;(, V  —  1 )  + to  /,  terms  }, 

n-j  ■ 

or,  with  ordinary  prices, 

« 

(A  formula  for  it  somewhat  like  the  last,  but  not  specifying  the  period  at 
which  the  weights  are  chosen,  is  given  by  Edgeworth,  B.  59,  j).  265. )  It  is 
confined  to  custom-house  returns,  and  so,  like  James's  method  already  (see 
next  section),  has  the  fault  of  representing  the  relative  weights  only  of  goods 
exported  or  imported.  He  makes  one  calculation  each  for  exports  and  for 
imports.  His  Aveights,  which  are  according  to  the  total  money-values  of  the 
classes,  are  reduced  to  percentages  of  the  total  money-value  of  all  the  exports, 
or  imports,  not  at  the  basic  period,  1861,  nor  on  an  average,  but  at  another 
period,  1875,  the  seventh  in  his  series  (which  skips  every  other  year).    These 


rtl^ii,  APPENDIX    C 

percentages  should  add  u\>  to  100  ;  but  as  he  could  not  niana,ii;e  all  the  classes, 
his  weights  for  exports  in  that  year  added  up  only  to  73.1  and  those  for  im- 
ports only  to  84.38  ;  wlierefore  these  figures  are  taken  instead  of  100  for  his 
starting  points  (in  1861,  the  prices  then  being  reduced  to  unity).  [He  said 
it  is  unimportant  of  what  year  he  chose  the  weighting,  because  the  totals  of 
the  percentages  varied  very  slightly — between  72.7  and  80.4  for  exports,  and 
between  82.42  and  88.88  for  imports.  B.  40,  p.  8,  cf.  B.  39,  p.  5.  This  is 
no  reason  at  all,  as  it  overlooks  that  the  particular  weights  of  the  individual 
classes  varied  considerably.]  In  extending  his  investigations  backwards,  in 
B.  36,  he  had  to  confine  himself  to  fewer  classes  still,  so  that  the  figures  (for 
the  weights  of  187o,  and  the  price  averages  of  1861)  reduce  to  65.8  for  ex- 
ports, and  to  81.16  for  imports.  (He  describes  his  raetliod  best  in  B.  45,  qq. 
709-716. ) 

?.  4.  Palgrave,  B.  77,  has  introduced  the  real  variation  of  employing  the 
weighting  of  every  subsequent  year  in  the  comparison  of  it  with  the  first  or 
basic  period  ;  and  he  appears  to  have  been  followed  by  Sauerbeck  in  one  of 
the  hitter's  "tests"  (see  B.  82,  p.  218).  Prices  are  taken  for  the  basic  period 
at  100  ;  l)ut  here,  for  simplicity,  we  may  take  them  at  1.00.  The  formula  for 
comparing  the  second  period  with  the  first  is 

P  1 

p?  =    -„  {di^a/  +  hi!^./  +  to  ?i  terms)  ; 

or,  adapted  to  the  prices  of  any  mass-units, 

?;==<(^^«i  +  ^v5;  + to  7^  terms), 


which  in  full  is 
P, 


^/  (  ■'■2«2  -  +  y-iih  ^  + to  71  terms  1  , 


(like  this  it  is  given  by  Edgeworth,  B.  59,  p.  265),  which  reduces  to 
P„        1     /     «.2           /i2   ,  \ 

P.=«/V^",    +^^^,    + to  «  terms  j. 

When  dealing  with  a  third,  or  any  subsecpient,  period,  Palgrave  compares 
it  with  the  first,  in  the  same  manner,  thus, 

P  1 

tT  =     //  ( a:i«/  +  b;i/:*:/  4- ion  terms ) .  ♦ 

Therefore  a  comparison  between  the  results  so  obtained,  as  representing  a 
comijarison  between  the  subsequent  periods  themselves,  means  a  comparison 
in  this  form, 

P        ,  //  (as'V  +  ba;V  4- to  ;i  terms) 

_  _  .^ , 

^  „{^i"'i  +"^2^2'  + to  n  terms) 

which,  by  restoring  the  values  of  n./'  and  n/',  and  converting,  reduces  to 

Ps  ^  a.,"/  +  ^a■Ji^^  -I-  . . ^. .    aj  +  ba  + 

P,    "  a,«./  +  ^a■Jl/  +  ■■  •  •  •  ■  a,  -f  bs  + ' 


REVIKW     OK    MF/l'lloDS  539 

and  the  same  may  be  more  generally  stated  thus, 

Is Qi         1^1 a^  +  D2 -f- _ 

^2  ~  o  "2  I  1,  1^^     ~  '  a,  +  bs  + ■ ' 

or,  lastly,  in  this  form, 

Pj  «2'    ,         /^2'    ,  '  %«:.  +  2/3/^3  + ■ 

•'•2  7^-1-2/273         i      

The  inverse  of  this  formula  gives 

,."2'.     'V,  

1/02  ~  X^S  +  2/2^2  + ■    ,    «3'     ,      „    /3s'     , 

^^«, +^^i,  + 

which  we  recognize  as  a  formula  with  double  weighting  (with  very  curious 
weighting  in  the  separate  averages  at  each  period). 

IV.    SCROPE'S   METHOD   AXD  ITS   VARIETIES. 

^  1.  Scrope,  1833  ;  see  B.  9. 

Tlie  method  there  described  may  ])e  thus  formulated, 

p         -,  (■'■«2  +  2//^2  + to  n  terms ) 

fj  '^^ 

P    ^  1  ' 

'        ^(xaj  +  y/3j  + ton  terms) 

this  representing  the  variation  in  the  "  mean  "  ;  or  "  in  the  sum  "  as  follows, 

Pz^xog  +  y/j^  +  

Pi       xfli  +  y/:(i+ • 

Scrope  did  not  specify  at  which  period,  or  how,  the  raass-([uantities  are  to 
be  chosen.     All  he  had  in  mind  was  a  general  average  of  some  sort. 

We  know  that  tliis  method  is  the  same  as  the  arithmetic  average  of  the 

price  variations  with  weighting  according  to  .mj,  l/|■t^,  (Appendix  A,  II, 

§  8),  and  as  the  harmonic  average  of  the  price  variations  with  weighting  ac- 
cording to  xflj,  yj3r,,  (Appendix  A,  IV.  ^3),  and,  in  some  cases,  as  the 

geometric  average  of  the  price  variations  with  weighting  according  to  the 
geometric  means  between  these  weights  (Appendix  A,  VI.  ^?7-8).  As  it 
virtually  employs  arithmetic  averages  of  the  prices  at  each  period,  it  is  simpler 
to  regard  it  as  the  arithmetic  average  of  the  price  variations  with  the  weight- 
ing of  the  first  period,  and  it  could  be  expressed  tlius,  in  the  com})arison  of 
the  second  period  with  the  first, 

t  =  ^(*^a;  +  ^';^;+  to«terms). 

In  this  form  it  might  be  regarded  as  a  variety  of  Young's  method,  and  it  was 


540  APPENDIX    C 

originally  no  doubt  suggested  without  an  idi'a  of  its  being  otbor  than  Young's 
method.      But  it  is  essentially  diil'erent. 

Its  differentia  from  Young's  method  is  that  it  uses  mass-quantities,  Mith 
hidden  weighting  of  the  price  variations,  instead  of  direct  weighting  accord- 
ing to  total  money-values.  This  rests  on  the  difference  that  it  compares 
price  averages  (or  price  sums)  at  two  or  more  periods,  instead  of  directly 
averaging  price  variations  between  two  periods. 

But  just  as  Young's  method  uses  single  weighting  in  every  average  of  price 
variations,  so  .Scrope's  method  uses  the  same  mass-quantities  at  both  periods 
in  every  comparison. 

Used,  as  recommended  by  Scrope  himself,  in  a  series  of  years,  always  on 
the  same  mass-quantities,  if  the  total  sum  of  the  money-values  at  the  first 
period  is  reduced,  as  usually  is  the  case,  to  100,  then  the  others  are  to  be 
reduced  accordingly  merely  by  dividing  them  by  the  same  figure.  Suppose 
this  figure  is  /-.     Then  the  index-numbers  ( I )  for  a  series  of  years  are 

Ii  =  ~(-^«i  +  2/.'^i+    )  =  100, 

l3=-(.ra., -Ly/i,  +  ), 

and  so  on.  A  comparison  between  any  of  these  index-numbers  obviously 
yields  the  same  result  as  a  direct  comparison  between  the  corresponding 
])criods  (using  the  same  mass-quantities). 

This  method,  with  more  or  less  undefined  mass-quantities,  has  been  recom- 
mended and  formulated  by  Fauveau,  B.  54,  p.  356,  and  by  Walras  ( who 
calls  it  the  method  of  "the  multiple  standard"),  B.  69,  pp.  15-16,  B.  70, 
pp.  432,  468,  B.  71,  p.  131  ;  and  has  been  recommended  by  Newcomb,  B.  76, 
p.  211,  Andrews,  B.  107  and  108,  J.  A.  Smith,  B.  129,  pp.  27-29,  Pomeroy, 
B.  135,  p.  332,  and  Parsons,  B.  136,  pp.  134-136. 

?  2.  Definiteness  in  regard  to  the  mass-quantities  may  be  given  to  this 
method  in  several  ways,  among  which  three  have  been  noticed  ;  ( 1 )  by  em- 
ploying the  mass-(juantities  of  the  first  (or  basic)  period  ;  (2)  by  employing 
the  mass-quantities  of  the  other  periods  which  are  compared  with  the  first  (or 
basic)  period  ;  or  (3)  by  employing  some  mean  between  these  methods,  two 
such  having  been  suggested.  All  these  have  been  formulated,  most  of  them 
reconnnended,  and  some  of  them  ai)i)lied. 

Tiie  first  two  were  first  distinguished  and  fornndatc'il  by  Drolusch,  B.  29, 
pp.  30-39,  wiio  added  notice  also  of  one  form  of  the  third,  B.  31,  i)p.  423- 
425,  but  who  rejected  tiiem  all,  except  in  an  accidental  case  when  it  hajipens 
that  the  mass-tpiantities  are  the  same  at  both  the  periods  compared.  They 
have  all  been  similarly  condemned  by  Wicksell,  unless  it  accidentally  hap- 
pens that  this  method  tried  in  the  first  way  and  in  the  second  way  yields  the 
same  result,  B.  139,  pp.  11-12.  On  the  other  hand  all  three  were  consid- 
ered equally  good  by  Sidgwick,  who  therefore  regarded  none  as  authoritative, 
B.  56,  pp.  67-68,  (cf.  Edgeworth,  B.  59,  p.  264).     Similarly  Padan,  B.  141. 


REVIEW    OF    METHODS  541 

]>ut  Lindsay,  likewise  considering  idl  tiiree  e(iually  <,r(K)d,  and    wantinfij  only 
one  to  be  employed,  is  indifVerent  which  he  adopted,  J{.  114,  [)p.  25-26. 

(1)  The  method  with  the  mass-(juantities  of  the  tii-st  or  hasic  period  was 
formulated  and  recommended  by  La.s})eyres,  B.  26,  p.  306  (who  seems  to  have 
been  unaware  that  Drol)isch  had  already  formulated  and  condemned  it). 
Adapted,  and  modified  (by  omittint;-  use  of  100),  his  formula  is 


P2  ^  ^iSjfli/Vti 
Pi       •'•i«i  +  Uilh  + 


In  a  series  of  years  later  comparisons  would  be  in  this  form, 
P3  _  ^.-ja,  +  y,/3,  + 


P2       aJifl^  +  yyi^  + 


in  which  the  weighting  of  the  price  variations  (arithmetically  averaged)  is 
rather  curious,  being  according  to  the  money-values  of  the  mass-cjuantities  of 
the  first  or  basic  period  at  the  prices  of  the  earlier  of  the  two  periods  com- 
pared. 

This  variety  of  the  method  has  been  used  by  Bourne,  in  one  of  his  opera- 
ations  upon  custom-house  returns,  B.  47-49.  Tlie  next  variety,  as  used  by 
Sauerbeck  for  periods  prior  to  his  basic  period,  is  like  this  in  using  the  mass- 
quantities  of  the  earlier  periods  (but  not  those  of  the  basic  period),  (cf.  Edge- 
worth,  B.  59,  p.  264). 

(2)  The  method  with  the  mass-quantities  of  the  later  periods  (or,  more 
generally,  of  the  periods  other  than  the  basic)  was  first  consciously  employed 
by  Paasche,  B.  33,  pp.  171-173,  who  has  been  followed  by  v.  d.  Borght,  B. 
55,  and  ('onrad,  B.  96  and  97.  The  same  method  has  also  been  employed, 
in  a  very  general  way,  and  dogmatically,  by  Mulhall,  B.  74,  p.  1157,  B.  75, 
p.  1  ;  and  in  his  second  "test"  by  Sauerbeck,  B.  82-90,  who  is,  partly,  fol- 
lowed by  Powers,  B.  131,  see  p.  29. 

The  formula  for  the  method  in  this  form  is 


P2  ^2«2  +  2/2/^2  + 


Pi        a:2«i  +  2/2^1  + 


(cf.  Edgeworth,  B.  59,  p.  264).  Thus  the  weighting  in  (arithmetically)  aver- 
aging the  price  variations  is  according  to  the  money-values  of  the  mass-quan- 
tities of  the  other  period  at  the  prices  of  the  first  or  basic  period. 

In  a  series  of  years,  a  comparison  between  any  two  of  the  later  years  would 
be  as  follows, 

a;s«3  +  yiA+ 


P3    _  ^3^lJ_^3^VL 

P2       a^2«2  +  2/2/^2  + 


^2«i  -*  y-iPi  + 

~  a;2«2  +  J/'A  + a^jfli -1- 2/3/^1   .   ' 

hich  uses  double  weigliting  in  a  curious  form  (the  weighting  of  the  average 
at  each  period  compared  being  according  to  the   nioney-values  of  its   mass- 


w 


542  APPENDIX    C 

quantities  at  the  prices  of  the  basic  period).  If  the  prices  Imve  all  been  re- 
duced so  that  they  are  at  unity  (or  100)  at  the  first  period,  the  mass  quantities 
being  correspondingly  (inversely)  altered,  the  formula  becomes 

-P,^x/a/  +  y/,3/-^ •>/  +  ///+ 

Pa       x/a/  +  2///3/  + x/ +  rj./  + " 

This  method  had  already  been  employed  in  a  particular  application  to 
custom-house  returns,  in  England  by  James,  B.  10,  (his  method  being  based 
upon  the  peculiarity  that  British  exi)orts  were  officially  rated  always  at  the 
same  prices  at  which  they  had  been  originally  rated  in  1694,'  and  that  since 
1798  there  were  added  to  the  records  also  the  "declared  values"  according  to 
the  current  prices  set  upon  their  goods  by  the  merchants  themselves,  so  that 
his  weighting  for  each  year  was  according  to  the  money-values  of  its  mass- 
(juantities  at  the  prices  of  1694)  ;  and  in  France  (on  the  "official  values" 
based  on  the  prices  of  1826 and  the  "actual  values"  recorded  after  1847),  both 
for  imports  and  exports,  separately  and  combined,  in  one  jump  from  1826  to 
1847  and  thereafter  annually  to  1856,  by  Levasseur,  B.  18,  pp.  181-184,  188- 
192.  This  work  was  resumed  by  De  Foville,  B.  50,  who  continued  it  down 
to  1862,  when  the  "official  values"  on  the  old  prices  of  1826  were  discon- 
tinued. (Thereafter  the  French  Custom-house  every  year  first  published 
"provisional  values  "  consisting  of  this  year's  mass-quantities  at  the  last 
year's  prices,  and  later  the  "actual  values"  consisting  of  this  yeays  mass- 
quantities  at  this  year's  prices.  To  compare  the  latter  with  the  former  for 
every  year  is  always  to  do  the  same  operation  as  here  indicated  in  the  formula 

p 
for  p- ,  a  new  basis  being  employed  every  time.     De  Foville  continued  his 

series  in  this  way  down  to  1880,  stringing  them  out  in  a  series  following  upon 
the  previous  series. ) 

(3)  A  mean  between  the  first  two  varieties  may  be  by  merely  using  each 
separately  and  drawing  the  (evenly  weighted  arithmetic)  mean  between  their 
results.     The  operation  is  expressed  by  this  formula, 

P2  ^   1    /a'l"2  +  yA+ ,     ^2""2^  ^2/32+ \ 

P,        2  V.ri«j  +  y,/?,4- -t- ^^„^  +  y^^^,  + )• 

This  appears  to  be  the  form  Drobisch  had  in  mind.  (It  is  so  understood  by 
Edgeworth,  B.  59,  p.  265.) 

Another  is  to  form  one  calculation  on  the  (evenly  weighted  arithmetic) 
mean  of  the  two  mass-quantities  of  every  class,  according  to  this  formula, 

P,  ^  ^(a:,-f  x,)a.,  +  j(yi  +  ?/2)/^2  + 

Pi        2l:'-i+^-2)«i  +  i(?/,  -l-y^)/^!-^ ■ 

This  formula,  which  reduces  to 

Xi  —  (•'•i  +  -r2)"=2-'-  iyi  +  y2)ih  '^    ••  •• 

Pi        (a;i+iC2)ai+(3/, -f-yji^Sj   +-  '• 

1  Or  Wm  or  ]f;!i7  ;  cf.  Flux,  B.  140,  p.  67,  who  has  recently  employed  the  same 
method. 


REVFEW    OF    METHODS  54,'} 

was  independently  suggested  by  Marshall  and  Edgeworth  (according  to  the 
latter,  B.  51,  p.  2(55),  and  is  reconiniended  by  the  latter,  ibiil.,  p.  200  (but 
not  decisively,  cf.  p.  255),  and  by  the  British  Association  Committee,  First 
Report,  B.  75,  pp.  249-250.  Lindsay  recommends  it  when  comparing  periods 
long  separated,  B.  114,  p.  25.  It  is  formulated  by  Zuckerkandl,  B.  115,  \>. 
248,  B.  116,  p.  247.  And  it  is  experimented  with  bv  Powers,  B.  131,  pp. 
28-29. 

As  regards  the  relationshij)  between  these  two  sub-varieties,  it  is  obvious 
that  if  in  the  formula  for  the  iirst  the  denominators  liappen  to  be  eciual  as 
wholes,  this  formula  reduces  to  the  formula  for  the  second  ;  but  not  otiierwise. 

The  arithmetic  average  of  the  mass-tpiantities  over  the  whole  epoch  in- 
vestigated (in  his  case  of  thirty  five  years)  lias  been  recommended  as  the  best, 
and  employed,  by  Powers,  B.  131,  p.  24. 

Another  mean  of  the  two  mass-quantities  of  every  class  is  the  evenly 
weighted  geometric.  Scrope's  method  applied  to  such  mean  mass-quantities 
is  recommended  in  this  work  (Chapt.  XII.)  under  the  name  of  Scrope's 
emended  method.     Its  formula  is 

Pi       ttjl/a:,!-,  -I-  yiVy^y^  +  

^  3.  It  may  here  be  added  that  it  would  be  obviously  absurd  to  attempt  to 
use  the  mass-quantities  at  both  periods  in  the  following  way, 

Pg  ^  Xi('2  +  y2l^2-\- 

Pi       ^-i^i  +  2/i/3i  + ' 

for  this  would  be  merely  a  comparison  between  the  total  valuations  or  inven- 
tories at  the  two  periods,  which  might  be  altered  by  a  change  in  the  mass- 
quantities  without  any  price  variations.  But  if  we  suppose  that  the  mass- 
quantities  have  not  changed,  i.  e.,  that  Xj  =  Xj,  y.^  =  j/j,  and  so  on,  this  formula 
would  be  the  same  as  any  of  the  preceding.  On  the  supposition  that  the 
mass-quantities  have  remained  nearhj  the  same,  it  is  employed  by  Edgeworth, 
B.  59,  p.  272,  cf.  pp.  264,  293  ;  also,  inverted  for  the  exchange-value  of 
money,  by  Nicholson,  see  here  below,  V.  ^  3.  Furthermore,  if  the  mass- 
quantities  all  change  alike,  so  that    ^  =    ^= = ''>  this  formula  would 

also  be  serviceable  if  we  multiply  the  denominator  by  /•,  in  order  to  counter- 
act the  change  of  the  mass-quantities  ;  for  then 

P,  ^       ->-2«2  +  3/2/32  + 

P,       /•(x,ai  +  2/A  + ) 

reduces  to 

P2  ^  Xggg  -I-  3/2/^2  + 

Pi      2-,fli  +  2/.A+  ' 

or  if  we  multiplv  the  numerator  bv   -  ;  for  then  it  reduces  to 

•    J. 

P2  ^  Xi«2-(-  y-A+  

P,      Ji«i  +  yA4- ' 


544  APPENDIX    C 

and  these  two  reduced  expressions  are  equal,  having  really  been  reached  in 

the  same  way,  namely  by   multiplying  the  original  expression  by  -  .     Cf. 

Appendix  A,  VII.  ^  5.     These  properties  have  been  made  use  of  by  Nichol- 
son, see  below  V.  ^  3. 

On   the   same   two   suppositions    the   following   formula    (of    Drobisch's 
method)  would  be  serviceable, 


—,  (^2«2  +  y-i^i  +  to  n  terms) 


P,_ 

P  ~   1  ' 

^  >  (^'I'^i  +  2/1/^1  ~t~ to  ft  terms) 

for,  in  the  second,  more  comijrehensive,  case,  this  would  reduce  to 

P.^n/  (-^-^^"+^2^^+ to  n  terms)  ^^^^^^^^^^^ 

Pi       — ,  (x,«i  +  2/A+ to  «  terms)       "^^'^^  +  ^^^^^^  + 


or  to 

P.   ,   .i>(^i°'  +  ^'^^  + ^""^^^"^^^^..«2  +  yA+ ■ 

Pi       J_  (:,^a^4_y^,3^+ to  n  terms)       ^i^^i  +  2/A  ^ ' 

ft, 

hut  not  otherwise. 

Thus  in  cases  when  the  mass-quantities  are  the  same,  or  proportional,  at 
both  or  all  the  {)eriods  compared,  all  the  varieties  of  the  method  reviewed  in 
this  Section  (and  some  of  the  methods  to  be  reviewed  in  the  next)  are  the 
same.  ^ 

Now  we  know  (see  Chapt.  XI.  Sect.  III.  ^^2-3)  that  on  the  supposi- 
tions here  made,  namely  of  the  mass-quantities  being  the  same,  or  propor- 
tional, at  both  or  all  the  periods  compared,  Scrope's  method,  applied  to  the 
same  mass-quantities,  or  to  the  propoi'tional  ones  at  either  period,  or  to  any 
mean  or  average  between  them,  is  perfectly  correct.  This  has  been  recog- 
nized, not  only  by  Edgeworth  and  Nicholson,  as  above,  but  also  by  Drobisch, 
B.  29,  pp.  30,  37,  B.  30,  pp.  147-148  ;  Sidgwich,  B.  56,  pp.  6G-67  ;  Lehr,  B. 
68,  p.  40;  Zuckerkandl,  B.  115,  p.  247,  B.  116,  pp.  244-245;  Wicksell,  B. 
139,  pp.  8-9  ;  Padan,  B.  141. 

V.     .METHODS   E]\rPLOYING  DOUBLE   WEIGHTING. 
\\.  Drobisch's  Method.     Drobisch,   1871  ;  see  B.  29  and  30    (31  being 
apologetic). 

He  always  considered  the  variation  of  the  arithmetic  averages  to  be  this, 

Z~iX^y^X—-  '  "''''^  "^^^^  ^«i  =  2//^i  = ( I^-  29,  p.  36,  cf.  pp.  31,  39) 

— i.  e.,  onlv  when  this  expression  is  equivalent  to  -(   ^  +  ^'^  +  )  ,  (see 

2  This  is  not  recognized  by  Powers,  B.  131,  pp.  27-28,  where,  by  Hiulty  arith- 
metic, he  divides  into  two  distinct  methods  what  is  really,  in  the  case  supposed, 
only  one  method. 


KKVIKW     OF    MF/nioDS  545 

Appendix  A,  II.  ?9).  In  other  words,  lie  always  took  "arithmetic average" 
(applied  to  price  variations)  to  mean  only  arithmetic  averaj^e  with  even  weight- 
ing. j\.nd  then  he  rightly  objected  to  tlie  general  use  of  the  arithmetic  aver- 
age so  understood,  B.  29,  p.  32,  and  rightly  condemned  Laspeyres  for  using 
it,  B.  30,  p.  145,  this  average  being  correct  only  in  the  particular  case  when 
the  mass-tjuantities  (numbers  of  the  mass-units)  happen  to  be  in  inverse  pro- 
portion to  the  prices  (of  the  same  mass-units)  at  the  first  period.  His  objec- 
tion, evidently,  was  only  against  the  use  of  even  weighting  in  averaging  price 
variations  ;  but  he  thought  it  was  against  the  use  of  the  arithmetic  average  in 
general.^  He  also  objected  to  the  method  of  using  uneven  weighting  when 
only  the  mass-quantities  at  one  of  the  periods  are  considered,  the  change  in 
the  mass-quantities  at  the  other  period  being  disregarded,  and  also  to  the 
method  of  using  the  mean  between  the  two  methods  (or  the  mean  of  the  mass- 
quantities?),  all  these  methods  giving  different  resulis,  none  better  than 
another,  B.  29,  pp.  36-39,  B.  31,  pp.  423-425.  He  thought  that  "the  fol- 
lowing consideration  does  away  with  all  difficulties,"  B.  29,  p.  39. 

Taking  his  prices  always  to  be  the  prices  of  the  same  mass-unit  (preferably 
a  hundredweiglit)  of  every  class  of  goods,  and  consequently  his  mass-quanti- 
ties to  be  the  numbers  of  these  mass-units  of  all  the  classes  of  goods  marketed 
at  the  place  in  question  during  the  period  in  question,  B.  29,  p.  34,  B.  30,  p. 

148,  he  represented  the  total  mass-quantities  of  the  classes  [A],  [B],  

marketed  at  the  first  period  by  x,  -f-  j/^  -j- ,  and  the  sums  spent  on  them 

by  ^lOi -f- j/i/?!  -|-  ■  ■'■■  ;  and  hereupon  he  asserted  that  the  average  price  of 

the  mass-unit  is  not  -  (  ^^"^  +  ^—  -f |,     not  -  (a.  +  J,  -f- )    ,  but 

n  \   )\  y,  /     L        w  J 

-— '— ,-^^^~i ,    (and  remarked  that  even  here  he  was  not  using  the 

■''■1  +  2/1  + 

arithmetic  average,  and  concluded  that  his  method  does  not  use  the  arith- 
metic average  at  all,  any  more  than  it  uses  the  harmonic  or  the  geometric, 
but  only  the  rule-of-three,  B.  29,  p.  40).  Proceeding  in  the  same  way  for  the 
second  period,  he  represented  the  then  average  price  of  a  mass-unit  of  goods 

by  AjLT^y2/jo^ _       Therefore,   neglecting   anv  average  of    the    price 

2:2  +  2/2  + 

variations,  he  represented  the  variation  of  the  price-average  by  the  varia- 
tion of  the  average  price  of  the  common  mass-unit  of  goods,  thus, 

^■i       X2  +  y-i  ^ 

1*1       X\'h  +  2/1'^  -t- 

•Ti    +  2/1     -I-    

^  3-2"'2  +  2/2/^2  + .   •'•1   +2/1   +    . 

^i«i  +  Vi^i  + '  •'■2  +  2/2  + ' 

^  He  also  noticed  that  if  the  mass-quantities  happen  to  be  in  inverse  propor- 
tions to  the  prices  at  the  second  period,  the  above  expression  reduces  to  the  .simple 
harmonic  average  of  the  price  variations  (cf.  Appendix  A,  IV.  ?  2)  ;  and  con- 
cluded that  even  on  the  assumption  of  the  mass-quantities  being  the  same  at 
both  periods,  the  "  arithmetic  average  "  is  no  better  than  the  "harmonic  average," 
and  vice  versd,  each  being  proper  in  certain  circumstances  and  improper  in  all 
35 


546  AIM'KNDIX    C 

and  the  variation  of  the  excliange-vahio  ()f  money  {Gcldiirrt) ,  he  added,  is 
the  inverse  of  this,  B.  2!),  p.  39,  B.  30,  pp.  148-U9. 

We  see  that  this  is  nothing  but  a  method  using  the  arithmetic  average 
with  different  numbers  of  figures  at  each  period—/,  e.,  witii  double  weigliting 
(cf.  Appendix  A,  VII.  ^1).  Tlie  arithmetic  average  used  by  Drobisch, 
when  he  is  drawing  the  average  price  of  the  mass-unit  of  all  goods  at  each 
period  separately,  is  the  least  imperfect  distinct  average  (for  each  period  sep- 
arately) that  he  could  draw,  since  it  takes  into  account  the  numbers  of  times 
the  mass-unit  is  repeated  in  every  class,  while  the  simple  average,  with  even 
weighting  of  the  classes,  in  tlie  form  rejected  by  him,  would  obviously 
be  imperfect.  But  the  fact  tliat  the  average  he  rejected  is  an  arithmetic 
average,  does  not  prevent  the  average  he  adopted  from  being  an  arithmetic 
average.^  He  thought  he  disproved  the  arithmetic  and  harmonic,  and  also 
the  geometric,  averages,  and  proved  that  a  method  using  none  of  them  is  the 
right  one.  As  a  matter  of  fact,  his  arguments  never  touch  upon  the  question 
as  to  which  kind  of  average  is  the  correct  one,  but  only  show  that  all  of 
them  are  wrong  with  even  weighting — which  everybody  admits.  And  he  re- 
adopted  the  arithmetic  average  without  any  argument,  but  assumed  it.  His 
attention  was  really  concentrated  upon  the  question  of  weighting,  and  it  was 
here  that  he  made  advance. 

In  this  method  it  is  material  that  the  mass-unit  should  be  the  same  in  all 
the  classes  of  things.  For,  although  in  the  numerator  for  the  average  at  each 
period  every  term  [e.  (/.,  J\a.^)  would  be  the  same  whatever  mass-units  are 
employed,  yet  in  the  denominator  the  corresponding  term  {e.  g.,  x^\  would 
})e  larger  or  smaller  according  as  the  mass-unit  is  smaller  or  larger.^ 

The  peculiarity  of  Drobisch's  metliod,  then,  is  that  it  uses  the  arithmetic 
average  (1  )  with  double  weighting  (2)  on  the  same  mass-unit  for  all  the 
classes. 

This  metiiod  is  praised  by  Roscher,  B.  32,  p.  275  ;  criticized  by  Laspeyres 
(see  here  Chapt.  V.  Sect.  VI.  ^5,  Note  11),  by  Paasche  (see  here  Cliapt. 
IV.  Sect.  V.  §1),  and  by  Lehr  and  Zuckerkandl  (see  here  Chapt.  V.  Sect. 
VI.  I  5,  Note  15)  ;  noticed  by  Edgeworth,  B.  59,  p.  265  ;  and  reviewed  by 
Lindsay  (who  said  it  rested  on  the  harmonic  mean!),  B.  114,  pp.  18-21. 
It  has  even  been  used,  in  application  to  goods  commonly  measured  in  the 

others,  B.  29,  pp.  .36-37 — a  conclusion  perfectly  correct  concerning  these  averages 
of  the  price  variations,  each  with  even-weighting,  but  without  any  application 
whatever  against  either  with  its  proper  weighting. 

2  Like  Drobisch  in  condeuniing  tiic  "arithmetic  average"  while  using  it 
themselves  (their  objection  being  really  against  only  the  arithmetic  average  of 
the  price  variations  with  even  weighting)  are  Geyer,  B.  28,  p.  321,  and  Paasche, 
B.  33,  p.  171,  cf.  p.  172,  B.  34,  p.  (K),  cf.  p.  64.  Even  Walras  does  not  appear  to 
perceive  that  the  fornuila  for  Scropc's  method  (using  tlie  same  mass  quantities  at 
each  period)  is  one  employing  arithmetic  averages  (and  is  connected  with  the 
arithmetic  average  of  price  variations),  sometimes  contrasting  it  with  the  geo- 
metric, the  harmonic  and  the  arithmetic  averages  of  the  price  variations  (always 
with  even  weighting),  as  if  these  alone  were  averages.  For  the  references,  see 
above  in  IV.  ?  1  end. 

3  Drobisch's  nicthud  is  tlie  only  one  yet  invented  in  which  this  feature  is 
essential. 


KKViKW    OF  MirnioDs  547 

same  mass-unit,  witliout  knowledge  of  Drohisli,  by  Powers,  B.  131,  see  j).  29. 

§2.  Lehr's  method.     Lelir,  1885  ;  see  B.  68. 

Starting  out,  like  Drobisch,  by  supposing  the  prices  of  all  articles  to  be  ex- 

j)ressed  on  the  same  mass-unit  (by  weight),  although  the  rest  of  his  method 

makes  this  unnecessary,  Lehr  notices  that  over  the  two  periods  togetlier  there  is 

spent  the  sum  of  (x^n^  -\-  oc^a^)  money-units  for  (a-j  -f  v^)  mass-units  of  the  class 

[A],  wherefore  the  average  price  of  one  mass-unit  of  [A]  for  both  the  periods 

.    x^a,  -\-  a'2«2  -11  •  , 

IS , money-unit,  and  the  raonev-unit  purchases  on  the  average  dur- 

]\  -\-  X.,  ^  I  .1  n 

ing  both  the  periods  ■  "  ^        -—  mass-unit  of   [A].     And  similarly  for  the 

article    [B]    over  the   two   periods    together,    there    is   spent    the   sum   of 

(.Vi/'^i  +  ^2/^2 )   money-units  for  (y, +  .'/2)    mass-units,    wherefore  the   average 

price  of  one  mass-unit  of  [B]  for  both  the  ijeriods  is  ^^^ — ^^^  money-unit, 

y\  +  2/2 
and   the   money-unit   purchases   on   the   average   during  both   the  periods 

}    ,       r,    mass-unit.     And  so  on  with  the  other   classes.     Now  in  every 

?/l/^l  +  2/2/^2  ^ 

class  the  mass-quantity  which  the  money-unit  purchases  on  the  average  over 
the  two  periods  together — or  the  mass-quantity  of  which  the  average  price 
during  the  two  periods  together  is  one  money-unit — he  calls  a  "pleasure- 
unit,"  p.  38,  having  first  defined  a  pleasure-unit  to  be  an  equivalent  quantity 
of  any  goods,  pp.  37-38.     The  total  quantity  of  [A]  which  comes  into  trade 

during  the  first  period  consists  of  v or  x,  I    '  '   , — -— ^  )  such  pleas- 

^"i  +  -t-.^  V     A-i  +  a:2     / 

ure-units  ;   and  the  total  quantity  of  [A]  which  comes  into  trade  during  the 

second  period  consists  of  X2  (  ^J—     ^  -  j  such  pleasure-units.    Similarly  the 

total  quantities  of  [B]  coming  into  trade  during  each  of  the  periods  consist 

of  yif^^'t^')   and   ,jJy^^y^^-A   such  pleasure-units  respectivelv. 

\     111    •    ifi      /  V      2/1    I    ^2      / 

And  so  on.     During  the  first  period,  then,  there  come  into  trade 

^■1  _r  +  Ih     -"'^'jf^^  H- to  ;/  terms 

pleasure-units,  for  which  are  paid  {xiO^-^  yyi^-\- ton  terms)  money- 
units  ;  wherefore  the  pleasure-units  cost  on  the  average,  or  their  average  price, 
expressed  in  money-units,  is 

p ^i«L+  2/1^1  +  •••: •• 


^.   /  -t'l^  4- ^2«2  \^,,   ( 2/1/^1  +  Vil^i  \    ,    ■ 

'  \     ■»•!  +  -r.,     /        •  '  V     2/1  +  2/2     / 

And  during  the  second  i)erio(l  there  come  into  trade 
\     •'•1  +  i'-i     )       "  \     2/1  +  2/2     / 


54S  AIM'K.NDIX    C 

pleasure-units,  costing  {x^a^-^  y^,3^+ to  h  terms)  money -units  ;  where- 
fore 


/  .r,fl,  +  a-jrtj  \  ,  „  /  yA+y-A  \ 


+ 


Therefore  the  variation  of  tliese  average  prices  is  the  quotient  of  the  expres- 
sion for  Pj  divided  by  the  expression  for  Pj,  which  reduces  to  this, 


P2  _Jvv+iA4^: 

Pi  "  i\^i  +  yA  + r  ( ^"iSj+^s  \  4_ , "  (  yA  -+  y^P 


( s'l^z  +  a;2«2  \  t„  (  yA  -^  ^2/^2  \ 


+     '■■ 


and  the  reciprocal  of  this,  Lehr  adds,  gives  the  variation  of  the  exchange- 
value  of  money,  pp.  38-40. 

Comparing  this  formula  with  Drobisch's,  we  see  that  Lehr's  formula  differs 
from  Drobisch's  only  in  the  second  half,  and  there  by  his  multiplying  both 
Xi  and  Tj  by  a  certain  expression,  Avhich  is  the  average  price  of  the  mass-unit 
of  [A]  during  both  the  periods,  and  by  multiplying  both  ?/,  and  1/2  by  anotiier 
expression,  which  is  the  average  price  of  a  mass-unit  of  [B]  during  both  the 
periods,  and  so  on,  in  every  instance  the  average  being  the  arithmetic  aver- 
age with  weighting  according  to  the  mass-quantities  of  each  period. 

Hence  the  peculiarity  of  Lehr's  method  is  that  it  uses  the  arithmetic  aver- 
age (1)  with  double  weighting  (2)  on  mass-units  that  have  the  same  (un- 
evenly weighted  arithmetic)  average  price  over  both  the  periods  compared. 

Tiuit  the  actual  mass-unit  used  in  every  class  is  inditlerent  is  plain  from 
the  fact  that  not  only  the  terms,  e.  g.,  x^fi-i,  in  the  first  half,  but  also  in  the 

second  lialt  tlie  terms,  e.  (j.,  x,  I 1'  luive  tlie  same  value  wliatever 

\     *i  ~r  ^2     /  * 

be  the  mass-unit  whose  price  is  a  and  whose  number  is  x. 

This  method  is  reviewed  by  Edgeworth,  B.  59,  pp.  265-266  ;  declared  the 
best  by  Zuckerkandl,  B.  115,  p.  248,  B.  116,  p.  248,  and  Wiebe,  B.  124,  p. 
165  ;  and  by  Lindsay,  B.  114,  pp.  22-24,  criticised  for  seeking  to  measure 
variations  in  the  mass-quantities !  Pointing  out  the  error  in  this  criticism, 
Wicksell  tries  to  refute  the  method  by  a  test  case,  which  supposes  the  j)rices  of 
ci-rtain  articles  to  be  zero  at  the  one  or  the  other  period  [and  which  therefore 
constitutes  a  case  in  which  these  articles  ought  to  be  excluded  from  the 
measurement],  B.  1.39,  pp.  lO-H. 

^  3.  Nicholson's  method.     Nicholson,  1887  ;  see  B.  94. 
Independently,  and  seemingly  without  knowledge  of  Drobisch's  metiiod, 
or  of  Lehr's,   Nicholson   has  invented  a  method   which,   being   somewhat 
vaguely  conceived,  turns  out,  on  analysis,  to  be  either  Drobisch's  method  nni- 
tiiated,  or  an  imperfect  fprm  of  Scrope's,  or  sometliing  else. 

Representing  the  aggregate  wealth  of  a  country  expressed  in  the  money- 
unit  (the  pound  sterling)  by  i>,  or  by  w,  he  sets  out  with  a  formula,  for  the 

first  period,  which  in  our  notation  is  this,  £,Wi  =  x^a^  +  y,/^j  + ;   wiiere- 

fore  tlie  purchasing  power  [or  exchange-value  in  all  other  things]  of  the 


i;i:vii;\v  of  mkthods  049 

money-unit  is  ,       o     ,  ,  PP-    307-308.      Similarly  at   the   second 

period,  on  the  supposition  that  the  mass-quantities  are  ahout  the  same  as 
before,  and  no  new  kinds  of  articles  liave  been  added  or  subtracted,  we  should 
have  £2^*2  =  ^2^2  +  !/-if^2  + >  ^nd  the  purchasing  power  of  the  money-unit 

is  -—-^j-     01 •     Therefore   we   should   have    (representing   the   ex- 

^2^2    r  .'/2/'2    r 

change-value  of  money  in  all  other  things  by  3fo), 

1 

Mm  _ -iaS  4- ya/?..  4- 

Moi        ^1 

ai«i  +  2/A+ 

%«2  +  ^2/^2  + ■ 

This  he  represents  as     -' ,    and    says  that,   on    these   suppositions,    "the 

change  in  the  purchasing  power  of  the  £  is  equal  to  the  fraction     ',"   and 

calls  this  fraction  "  the  coefiicient  of  appreciation,"  pp.  .309-310.  The  same 
is  true,   he  further  says,  if  the   mass-quantities  have  increased  uniformly, 

which  uniform  increase  he  represents  bv  m,     wherefore  jh  =  -^  :^  —  = = 

L  -h      Vi 

^+12_4l 
a^i  +  2/1  + 

dard  will  obviously  be  the  same  for  the  old  inventory  and  for  every 
uniform  addition  to  it,"  so  that,  although  the  new  coefficient  of  appre- 
ciation for  these  cases  is    -  •  m  ,   vet  either  "  the  change  in  the  standard  may 

be  measured  by  the  old  inventory  at  the  old  prices  divided  by  the  old  in- 

,  .       ,,  r.  M02       X\a, -[- y,3,  -[- "I  „,^ 

inventorv  at  the  new  prices"  \  i.  e.,   r--—  =  -^-^       "''    ,  pp.  310- 

311,  or  "the  change  in  the  purchasing  power  of  the  standard  is  found  by 
dividing  the  value  of  the  new  inventory  at  the  old  prices  by  its  value  at  the 

new"  \i.  c,  ^  ="^"^t^f  t"    -T  P-  313.     [Cf.  above  IV.   ?  3.- 

Thus  far  we  have  merely  Scrope's  method  inverted,  because  Nicholson  is  deal- 
ing with  variations  in  the  excliange- value  of  money  in  all  other  tilings.]  All 
this  he  thinks  accurate  enough  when  we  are  dealing  with  periods  near  to- 
gether. His  new  method  lie  begins  when  dealing  with  periods  far  apart,  in 
which  the  classes  of  goods  increase  and  decrease,  and  old  ones  become  e.\tinct 
and  new  ones  come  in — and  all  deserve  to  be  allowed  for.  Here  he  again  lets 
m  represent  the  increase  in  the  mass  quantities.  He  is  now  very  vague  as  to 
how  m  is  to  be  estimated.  He  is  willing  to  estimate  it  "  generally,"  p.  312, 
or  "from  various  independent  considerations,"  ]).  316.  When  the  classes  are 
the  same  at  both  the  periods,  he  descril)es  the  estimation  more  minutely,  in  a 

i'„«,  -(-  y.jit  -\- 010  o-io 

wav  expressible  in  our  notation  thus,   m  =: ^^ — ; ,  pp.  312-313 

*  ritti  4-  .i/,/ii  -I- 


1-  -^1       yi 

,   on   the  ground   that  "the  purchasing  power  of  the  stan- 


550  APPENDIX    C 

(cf.  Hourne  in  one  of  his  methods,  B.  47-49).  When  the  classes  are  not  all 
tlie  same,  lie  says  "  m  should  be  proportioned  to  the  increase  in  the  popula- 
tion and  to  the  increase  in  the  efficiency  of  tlie  labor  and  capital,"  p.  316. 

Such  an  increase  would  seem  to  be  i-epresented  either  by   -~ — ^-^f 

^•i  +  2/i+ 

(provided  the  numbers  be  cxpi-essed  as  those  of  some  common  mass-unit, 
about  which  Nicholson  is  silent),  or,  again  supposing  the  classes  to  be  the 
same  at  both  the  periods,   the  increase  would  seem  to  be  represented    by 

(^.j_^_|_ j.      However  this  be,  the  new  coefficient  of  appreciation 

n  V  ^1       2/i  / 

we  find  still  to  be     ^  •  w  ,   which  cannot  be  reduced  now  as  it  could  be  be- 

"'2 

fore,  p.  315.  Hence,  the  variation  of  "general  prices"  being  tlie  inverse  of 
the  purchasing  power  of  money,  pp.  317,  318,  his  formula  is 

which,  in  full,  according  to  the  methods  of  estimating  m,  is  either 


P'i          •'*"2«2  +   2/2/^2   +    •  • 

■■■■    -''i"!  +  yA  H — 

n  \ 

Pi       ^i«i  +  2/A  +  -- 

•  •  ■  ■  "  ^2^1  +  2/2/3i  -\ — 

. . .  ' 

\^) 

Pa  _  x-jS  +  2/2/32  H 

••••    ^i  +  j/i  + 

(2) 

Pi       ^-i"!  +  2/i/3i  +  •  • 

■•••     X2  +  y2-r ' 

P2       x^a^  +  2/2/^2  +  •  • 
Pi       a;la^+^/^/3l  +  •• 

-•)• 

(3) 

The  first  reduces  to 

P2       ^i^i 

1  +  2/2/^2+ 

Pi     Vi-^y-z^i  +  -- 

which  is  Paasche's  variety  of  Scrope's  method.     It  is  plainly  no  better  than 
P2  _  -f2«2  +  ^2/^2+  ,  •gi«2  +2/1/32+ 

Pi    ~  ■'•l«l   +  2/l/3l   +      '    -|-2«2  +  2/2/32  + ' 

which  reduces  to 


ij  ^  •^•i«2  +  yA  + 

Pi      --Ki", +  2/i/3i~+ ' 

which  is  Laspeyres's  variety  of  Scrope's  method.  Hence  it  is  evident  that  in 
the  second  half  some  mean  must  be  employed  of  the  prices  at  the  two  periods. 

The  second  is  the  formula  for  Drobisch's  method.  But  in  this  form  Nich- 
olson's nietliod  is  not  the  same  as  Drobisch's,  since  Nicholson  does  not  specify 
how  the  mass-units  are  to  be  chosen.  In  this  form  this  method  is  wholly 
haphazard. 

The  third  is  the  most  original  of  the  three  forms.*     But  it  contains  a  de- 

"•A  variation  ajxni  it  niuy  be  suggested  in  the  followiuf^, 


Pi       ■••|«i  +  2/1,^1  + '  *^     a-/ 2/2 


REVIEW    OF    METHODS  551 

feet  in  averaging  the  inverted  variations  of  the  mass-f|uantities  with    even 

weigliting.'^ 

Which  of  these  three  tlie  method  really  is,   depends  upon  the  way  tlie 

calculations   have  heen  made  by  the  statisticians  upon  whom   Nicholson   is 

willing  to  rely.     For  a  peculiarity  in  his  method,  as  used  by  himself,  is  that 

he  makes  separate  use  of  the  two  halves  in  the  formula.     lie  accepts  some 

statistician's  rough  appraisal  of  the  comparative  wealth,  expressed  in  money, 

of  a  country  at  two  periods,  calculated,  e.  (/.,  from  the  income  tax,  p.  318, 

,  ,  .       .    .        j:Mo  +  u,/?,  -I- . 

(this  giving  —      ^^— — ,  although  no  particular  things  are  noticed, 

and  permanent  things,  such  as  real  estate  and  other  capital,  are  included,  if 
indeed  the  appraisal  is  not  principally  of  them)  ;  and  again  he  accepts  some 
other  statistician's  rough  estimate  of  the  comparative  wealth,  in  commodities, 
of  the  country  at  the  two  periods,  calculated,  e.  (j.,  from  the  increase  of  popu- 
lation and  improvements  in  machinery  and  methods  of  production,  ibid.,  (this 

giving,  by  inversion,  either  -^   ;         , ,  or  -  (  -  +  -  + ),   indef- 

a-i  +  yzH- nXx^      2/2  / 

initely,  without  notice  of  particular  things,  but  this  time  with  reference 
chiefly  to  products).  In  this  way  he  even  thought  he  could  show  that  be- 
tween 1848  and  1868  money  appreciated,  or  "general  prices  "  fell  !  pp.  318- 
320. 

This  method  lias  been  described  by  the  Gold  aud  Silver  Commission, 
Final  Report,  London  1888,  p.  23,  as  "totally  distinct"  from  the  usual 
methods.  Its  partial  similarity  with  Drobisch's  method  seems  to  have 
escaped  notice. 

^  4.  Other  methods  involving  double  weighting  we  have  found  to  be  acci- 
dentally produced,  in  the  comparison  of  later  periods  Avith  one  another,  by 
methods  using  single  weighting  in  the  comparisons  of  the  later  periods  with 
a  common  basic  period.  For  these  see  above,  III.  ^  4,  and  IV.  ?  2  (2).  The 
latter  of  these  may  be  described  as  a  method  the  peculiarity  of  which  is  that 
it  uses  the  arithmetic  average  (1)  with  double  weighting  (2)  on  mass-units 
that  have  the  same  price  at  some  other  period  chosen  as  base. 

A  method  suggested  in  this  work  in  Chapt.  XII.  Sect.  II.  ?  4,  Note  9,  as 
an  improvement  on  Lehr's  is 

Ej ^ ^•2<'2  +  y-A+ -^i ("i  +  «2^  + j/i(/^i  4-/^2)  + 

Pi     -^i"! -I- yi/3i  4 ■^2(«i  +  «2)+yi(/3i  +  /32)  + ' 

the  peculiarity  of  which  is  that  it  uses  the  arithmetic  average  (1)  with  double 
weighting  (2)  on  mass-units  that  have  the  same  (evenly  weighted  arithmetic) 
average  price  over  both  the  periods  compared. 

*An  improvement  may  therefore  be  suggested  as  follows, 

P2  ^  ^•2^2 +  2/2/^2-1 ■  •  •       1    /  „  ■''i    I   \.V\ 

Pi      •'•i«i  +  //AH — 

or  better  still, 


7,  {  a*-'  +b*''  + to  n  terms  ), 


T>  = \ o — i •  1/     1        )    ■  I        I    to  »  terms, 

in  which  a,  b,  represent  means  between  the  weights  of  the  two  periods- 


OO'i  AIM'KNDIX    (" 

A  method  recommended  in  this  work  in  (liapt.  XII.  Sect.  II.  is 

Pi       i-i«i  +  yA+ 2-2 /a-ia-.^  +  1/2 /y,y2  + 


the  peculiarity  of  which  is  that  it  uses  the  aritlimetic  average  ( 1 )  with  double 
weighting  (2)  on  mass-units  that  luive  the  same  (evenly  weighted  geometric) 
average  price  over  both  the  periods  compared. 


BIBI.IOGRAIMI  Y 

OF   WORKS   DEALING  WITH  THE  MEASUREMENT 

OF  THE   EXCHANGE-VALUE  OF  MONEY 

BY   COMPARING   MANY    PRICES. 

W.  Fleetwood 

1.  Chronicon  preciosum  :  or,  an  account  .  .  .  of  the  price  of  corn 
and  other  commodities  .  .  .  shewing  from  the  decrease  of 
the  value  of  money  .  .  .  that  a  Fellow,  who  has  an  estate  in 
land  of  inheritance,  or  a  perpetual  pension  of  five  pounds  per 
annum,  may  conscientiously  keep  his  Fellowship,  and  ought 
not  to  be  compelled  to  leave  the  same,  tho'  the  Statutes  of  his 
College  (founded  between  the  years  1440  and  1460)  did  then 
vacate  his  Fellowship  on  such  condition. — London  1707.  (2d 
ed.  1745.     See  especially  pp.  48-49,  135-137.) 

To  tind  the  number  of  i)ounds  which  have  the  same  exchange- 
value  as  the  five  pounds  formerly  had,  he  inquires  how  much  corn, 
meat,  drink  and  cloth  that  sum  would  then  purchase,  and  what  sum 
is  now  needed  to  purchase  them.  [His  proportions  are  so  nearly 
the  siime,  mostly  six  times,  that  he  escapes  the  question  of  aver- 
aging the  present  prices  of  the  mass-quantities  formerly  priced  at 
five  pounds,  and  also  the  question  of  weighting.] 

Dutot 

2.  Reflexions  politiques  sur  les  finances  et  le  commerce. — The 
Hague  1738, 18mo.,  Vol.  L,  pp.  365-377. 

Compares  the  total  sums  made  up,  at  two  periods  (times  of  Louis 
XII  and  Louis  XIV),  of  the  prices  of  the  customary  mass-units  of 
various  articles  (including  wages).  [Uses  the  arithmetic  average  of 
prices  with  haphazard  weighting.     See  Appendix  C,  1.] 

N.  F.  Dupr6  de  Saint-Maur 

3.  Essai  Hur  les  uujnnoies,  ou  reflexions  sur  le  rapport  entre  I'argent 
et  les  denrees. — Paris  1746,  4to.  (The  Reflexions  occupy  pp.  19- 
104.     See  also  pp.  5-6,  and  in  the  second  part  pp.  161-182.) 

Uses  many  desultory  price  notices,  but  principally  of  grain,  to 
compare  the  general  prices  of  his  day  with  those  in  the  period  be- 
fore the  discovery  of  America.      His  general  condusii)!!  is  an  average 

553 


504  hii;i,I(>(;i;ai'1IV 

of  some  sort,  unspecified.  But  he  uses  the  arithmetic  average,  with 
even  weighting,  in  drawing  tlie  average  price  of  single  articles  over 
many  years.     Criticizes  Dutot  for  his  data,  not  for  liis  method. 

G.  R.  Carli 

4.  Del  valore  e  della  proporzlone  de'  metalli  monetati  con  i  generi 
iu  Italia  prima  delle  scoperte  dell'  Indie  col  coufronto  del  valore 
e  della  proporzioue  de'  tempi  nostri. — 1764.  (Ed.  Custodi, 
Operescelte  di  Carli,  Vol.  I.,  pp.  299-366  ;  in  particular  §  IV., 
pp.  335-354.) 

Averaging  the  prices  of  grain,  wine  and  oil,  in  the  periods  about 
1500  and  1750,  he  compares  them  by  taking  the  earlier  as  units  and 
reducing  the  later  to  the  proper  figures  in  proportion,  thereby  repre- 
senting the  variations,  and  draws  the  aritliraetic  average,  [thus  using 
even  weighting.     See  Appendix  C,  II.]. 

G.  Shuckburgh  Evelyn 

5.  An  account  of  some  endeavors  to  ascertain  a  standard  of  weight 
and  measure.  (Philosophical  Transactions  of  the  Royal  Society 
of  London,  1798,  Part  I.,  Art.  VIII.,  4to.,  pp.  133-182.  Only 
pp.  175-176,  less  than  one  page  of  printed  matter,  besides  a 
table,  devoted  to  the  subject  of  measuring  exchange-value, Hhe 
rest  treating  of  weights  and  lengths.) 

Calculates  the  "depreciation  of  money"  from  1050  to  1800  by 
taking  the  prices  of  1550  at  100  and  reducing  the  prices  at  the  other 
dates  in  the  same  proportions.  Each  price-figure — for  Avheat, 
butcher's  meat,  day  labor,  and  twelve  other  agricultural  products 
lumped  together — is  counted  as  equally  important,  and  the  arithmetic 
average  is  drawn.  [Thus  even  weighting  is  used,  but  with  subordi- 
nation of  each  of  tlie  twelve  articles,  /.  e.  irregular  weighting  by 
classification.     See  Appendix  C,  II.,  §3.] 

Arthur  Young 

6.  An  en(iuirv  into  the  progressive  value  of  money  in  England  as 
marked  by  the  price  of  agricultural  products  :  with  observa- 
tions upon  Sir  G.  Shuckburgh's  Table  of  Appreciation  :  the 
whole  deduced  from  a  great  variety  of  authorities,  not  before 
collected. —  London  1812,  viii  pp.  and  pp.  66-135. 

Objects  mostly  to  Evelyn's  prices  and  autiiorities,  but  also  to  his 
method,  for  counting  every  article  as  equally  important.  Counting 
wheat  five  times  "on  account  of  its  importance"  by  value,  barley 
and  oats  twice,  provisions  four  times,  day  labor  five  times,  wool,  coal 
and  iron  each  once,  he  nmltiplies  the  percentages  of  the  variations 
betwen  two  periods  by  tiiese  weights,  adds  up  the  products,  and  di- 
vides by  nineteen,  the  sum  of  tlie  weigiits,  [thus  introducing  uneven 
^  All  this  has  been  reprinted  by  11.  Giffcn  in  the  Bulletin  de  I'lnstitut  interna- 
tional de  Statistique,  Rome  1H87,  pp.  i;32-l.'54. 


BIHLIO(JRAl'IIY  .555 

Aveighting,  with  the  arithmetic  average  of  the  price  variations.  See 
Appendix  (',  III.].  Hut  in  some  calculations  lie  also  uses  Evelyn's 
irregular  weighting  by  classification. 

J.  P.  Smith 

7.  The  elements  of  the  science  of  money.— London  1813.  (Ap- 
pendix, Art.  I.,  "  Estimate  of  the  effective  debasement  of  money 
in  the  eighteenth  century,"  pp.  471-476.) 

Applies  Evelyn's  [form  of  Carli's]  methoil  (with  strictly  even 
weighting),  to  the  reduced  prices  furnished  by  Young.  [V'ery  care- 
less. ] 

Joseph  Lowe 

8.  The  present  state  of  England  in  regard  to  agriculture,  trade, 
and  finance. — Loudon  1S22,  pp.  261-291,  and  Appendix,  pp. 
85-101. 

Seeking  to  measure  the  "power  of  money  in  purchase"  in  order 
that  debts  may  be  paid  in  the  same  value,  or  at  least  that  it  may 
serve  as  a  "table  of  reference,"  forming  a  "standard  from  ma- 
terials," he  adopts  Young's  method,  (but  omits  labor). 

G.  Poulett  Scrope 

9.  Principles  of  political  economy  .  .  .  applied  to  the  present 
state  of  Britain.— London  1833,  18mo.,  pp.  405-408. 

Following  Lowe  in  the  object  sought  (the  establishment  of  a 
" tabular  standard "  ),  suggests  this  method  :  "Take  a  price-current, 
containing  the  prices  of  one  hundred  articles  in  general  request, 
in  quantities  determined  by  the  proportionate  consumption  of  each 
article — and  estimated  ...  in  gold.  Any  variation  from  time  to 
time  in  the  sum  or  the  mean  of  these  prices  will  measure  .  .  .  the 
variations  which  have  occurred  in  the  general  exchangeable  value  of 
gold"  (p.  406).    [For  the  formulation  of  this  see  Appendix  C,  IV.] 

[Henry  James] 

10.  The  state  of  the  nation.  Causes  and  effects  of  the  rise  in  value 
of  property  and  commodities  from  the  year  1790  to  the  present 
time. — Loudon  1835,  pp.  12-15  and  a  table. 

Measures  the  rise  and  fall  in  value  [=  money-value,  price]  of 
British  produce,  from  1798  to  1823,  by  the  difference  between  the 
"official  values"  (always  on  the  prices  of  1694)  and  the  "declared 
'  values"  (according  to  current  prices)  of  British  exports.  [Applies 
Scrope's  method  to  custom-house  returns.  See  Appendix  (',  IV., 
?2(2).] 

G.  R.  Porter 

11.  The  progress  of  the  nation,  in  its  various  social  and  economical 
relations,  from  the  beginning  of  the  nineteenth  century. — Lon- 
don 1838.     (2d  ed.  1847,  pp.  439-440,  444-445.) 


556  lUm.KKiH.MMIV 

Employs  [Carli's]  metliod  on  inmuhly  prices  of  fifty  articles  in 
London  from  Jannary  1833  to  December  1837,  the  reduced  price- 
figures  being  carried  out  to  four  decimal  places.  Calls  the  unit-price 
started  with,  the  "index  price."  * 

M.  C.  Leber 

12.  Essai  sur  I'appreciation  de  la  fortune  privee  au  moyen  age. — 
Paris,  2d  ed.  1847,  335  pp. 

Roughly  measures  the  purchasing  power  of  silver  in  relation  to 
many  commodities,  etc.,  from  the  8th  and  11th  centuries  to  the 
present,  but  differently  for  the  poor  and  the  rich.  The  method  is 
not  exi)lained,  [but  is  probably  a  rough  arithmetic  average  of  prices, 
reduced  to  francs]. 

R.  H.  Walsh 

13.  Elementary  treatise  on  metallic  currency. — Dublin  1853,  pp. 
94-96. 

Quotes  and  reviews,  with  apparent  approval,  Scrope's  scheme  and 
method. 

A.  Soetbeer 

14.  Das  Gold.  (Die  Gegenwart,  Leipzig  1856.  Tables,  pp.  587- 
588.) 

15.  Ueber  die  Ermittelung  zutreffender  Durchschnittspreise. 
(Vierteljahrschrift  fur  Volkswisseuschaft  und  Kulturgeschichte, 
Berlin  1864,  Band  III.,  pp.  8-32.) 

16.  Materialien  zur  Erlaiiterung  und  Beurteilung  der  wirtschaft- 
lichen  Edelmetallverhaltnisse  und  der  Wahrungsfrage. — Berlin 
1886,  4to.,  pp.  81-117. 

In  the  first  gives  tables  of  prices  in  Hamburg  for  1831-40,  1841-50, 
1854,  1855,  and  of  the  same  reduced  to  100  at  the  first  period  ;  but 
does  not  add  or  average  them.  In  the  second  considers  only  the 
arithmetic  average.  In  the  third  employs  it  on  the  prices,  from 
1851  to  1885,  on  the  basis  1847-50,  of  one  hundred  Hamburg  and 
fourteen  British  articles,  arranged  in  different  divisions  in  which  the 
articles  are  evenly  weighted,  and  which,  in  the  final  averaging,  are 
evenly  weighted.      [Evelyn's  irregular  weighting  by  classification.] 

J.  Maclaren 

17.  Sketch  of  the  history  of  currency. — London  1858,  pp.  311- 
312. 

Reviews  Scrope's  scheme  and  method. 

^  Porter's  table  was  put  in  evidence  by  .T.  B.  Smith  l)efore  the  Select  Commit- 
tee on  Banks  of  Issue,  1840,  p.  31,  following  q.  362.  It  was  severely  condemned, 
for  the  use  of  even  weighting,  before  the  same  Committee,  q.  3()15,  by  Th.  Tooke, 
wlio  said  that  I'orter  Iiad  submitted  to  liiiu  lli(>  frame-work  of  his  tal)le  before 
publication. 


BIHl.IOCRAPIIV  557 

E.  Levasseur 

18.  La  question  de  Tor. — Paris  1858.  (Measurcraciits,  jip.  179- 
195.) 

Uses  partly  [Dutot's]  method,  and  partly  [.James's],  the  latter 
applied  to  similar  French  "  official  values"  (at  the  prices  of  1826) 
and  "actual  values,"  from  1847  to  1856.  [See  Appendix  C,  IV. 
^2(2).] 

W.  Newmarch 

19.  Mercantile  reports  of  the  character  and  results  of  the  trade  of 
the  United  Kingdom  during  the  year  1858 ;  with  i-eference  to 
the  progress  of  prices  1851-9.  (Journal  of  the  Statistical  So- 
ciety of  London,  Vol.  XXII.,  1859,  pp.  76-100.) 

20.  Results  of  the  trade  of  the  United  Kingdom  during  the  year 
1859  ;  with  statements  and  observations  relative  to  the  course 
of  prices  since  the  year  1841.  (Ibid.  Vol.  XXIII.,  ISGO,  pp. 
76-110.) 

21.  Commercial  history  and  review  of  1863.  (Supplement  to  the 
Economist,  Feb.  20th  1864,  folio,  pp.  4,  40-46.) 

In  the  first  reduces  the  prices  of  twenty  articles  in  the  year  1851 
to  100,  and  gives  their  proportionate  prices  for  the  years  following 
(omitting  '53,  '54,  '55,  '56).  In  the  second  reduces  the  prices  of 
twenty  two  articles  in  the  period  1845-50  to  100,  and  gives  their  pro- 
portionate prices  for  the  years  following  (with  the  same  omissions). 
In  the  third  continues  the  latter  operations,  and  liegins  the  series 
published  annually  in  the  Economist.  In  none  are  the  reduced 
prices  added  or  averaged.  ( The  addition  was  first  made  in  the  Sup- 
plement of  March  13th  1869,  where,  on  p.  44,  appeared  for  the  iirst 
time  the  "Total  Index  No.") 

W.  S.  Jevons 

22.  A  serious  fall  in  the  value  of  gold  ascertained,  and  its  social 
effects  set  forth. — London  1863.  (Republished  in  Investigations 
in  currency  and  finance^  London  1884,  pp.  15-118.) 

23.  The  variation  of  prices  and  the  value  of  the  currency  since 
1782.  (Journal  of  the  Statistical  Society  of  London,  June 
1865.     Republished  in  Investigations ^  pp.  119-150.) 

24.  The  depreciation  of  gold.     (Letter    in    the  Economist,   May 
.     gth  1869.     Republished  in  Investigations,  pp.  151-159.) 

In  the  first  raises  the  (juestion  whether  the  average  of  price  varia- 
tions should  be  the  arithmetic  or  the  geometric,  and  adopts  the  latter, 
which  he  always  uses  with  even  weighting.  The  table  contains 
thirty  nine  chief  articles,  from  1845  to  1862.  In  the  second  defends 
the  geometric  average  against  Laspeyres  (see  below  No.  25),  and 
introduces  consideration  also  of  the  harmonic  average.  The  table 
is  extended  to  cover  the  years  1782-1865,  on  many  articles  in  ten 


558  lUHLKKiUAiMIY 

divisions  [apparently  with  irregular  weighting  by  classification,  like 
Evelyn's]. 

E.  Laspeyres 

25.  Hamburger  Waarenpreise  1850-1863  und  die  californisch- 
australisclien  Goldentdeckungen  seit  1848.  Ein  Beitrag  zur 
Lclire  von  der  Geldentwerthung.  (Jahrbiicher  fiir  National- 
oekonomie  und  Statistik,  Jena  1864,  Band  III.,  pp.  81-118.) 

26.  Die  Berechnung  einer  mittleren  Waarenpreissteigerung.  (Ibid. 
1871,  Band  XVI.,  pp.  296-314.) 

Em])l()ys  [Carli's]  method  on  the  prices  of  forty  eight  articles,  the 
basis  being  1831-40,  and  coming  down  to  18G3.  Combats  Jevons's 
geometric  average  and  gives  an  argument  for  the  arithmetic,  in  the 
first  work.  In  the  second  defends  his  own  position  against  Drobisch 
(see  below  No.  30)  and  Geyer  (No.  28),  against  whose  methods  he 
delivers  a  counter-attack.  Admits  that  uneven  weighting  ought  to 
be  employed,  and  gives  an  emended  formula  [for  a  method  like 
Scrope's,  but  with  the  mass-quantities  specified  to  be  those  of  the 
first  period— one  of  the  formula? already  rejected  by  Drobisch  in  No. 
29]. 

A.  Walker 

27.  The  science  of  wealth. — Boston  1866.  (5th  ed.  revised  1869, 
pp.  177-178,  481,  488.) 

Employs  [Dutot's]  method  on  the  prices  of  ten  articles  in  New 
York  from  1834  to  1859,  of  seven  in  Calcutta  from  1850  to  1854  and 
from  1863  to  1867,  and  of  sixteen  in  Boston  from  1862  to  1865. 

Ph.  Geyer 

28.  Theorie  und  Praxis  des  Zettel-Bankwesens. — Munich  1867. 
(2d  ed.  1874,  Appendix,  pp.  321-326.) 

Condemns  Laspeyres' s  arithmetic  and  Jevons's  harmonic  {sic) 
averages,  and  then  reconstructs  a  method  very  much  like  Laspeyres' s, 
differing  only  in  reducing  the  first  prices  to  ahouf  2.00,  [/.  e.  only 
nominally,  and  with  slipshodness  ;  virtually  Carli's  method]. 

M.  W.  Drobisch 

29.  IJeber  Mittelgrossen  und  bie  Anw^endbarkeit  derselben  auf  die 
Berechnung  des  Steigens  und  Sinkeus  des  Geldwerths.  (Be- 
richte  iiber  die  Verhandlungen  der  Koniglich  siichsischen 
Gesellschaft  der  Wissenschat'ten  zu  Leipzig ;  Mathematisch- 
physlsche  Classe.     Band  XXIIL,  1871,  pp.  25-48.) 

30.  Ueber  die  Berechnung  der  Veriinderungen  der  Waarenpreise 
und  des  Geldwerthes.  (Jahrbiicher  fiir  Nat.-oekon.  und 
Statistik,  1871,  Band  XVI.,  pp.  143-156.) 

31.  Ueber  einige  Einwiirfe  gegen  die  in  diesen  Jahrbiiehern  ver- 
(iffentlichte  neue  Methode,   die  Veriinderungen    der  Waaren- 


iui5iJO(;i;Ai'in-  559 

.preise  und  des  Galdwerthes  zu  hertH-lincn.     {Ibid.  1871,  Band 
XVI.,  pp.  416-427.) 

Would  settle  the  di.si)iite  between  Laspeyres  and  Jevons  by  re- 
jecting both  the  arithmetic  and  the  geometric  averaging  of  price 
variations,  on  the  ground  that  the  ma.ss-quantities  ought  to  be 
taken  into  account  at  each  period.  Introduces  a  method  employing 
double  weigliting,  a  feature  in  which  is  that  tlie  mass  units  are  all 
the  same.  Thinks  he  rejects  all  averages,  making  use  only  of  the 
rule-of-tiiree.  [But  really  compares  the  arithmetic  average  of  the 
preciousness  of  commodities  at  each  period.  See  Appendix  C,  V. 
§  1,  and  Chapt.  V.  Sect.  VI,  §o.]  Himself  formulates  for  the  first 
time  [Scrope's]  method  applied  to  the  mass-quantities  of  the  first, 
and  of  the  second,  jieriod,  and  mentions  a  mean  between  the  two. 

W.  Roscher 

32.  Die  Grundlageu  der  Natioualoekonomie. — Stuttgart,  10th  ed. 
1873,  §  129,  pp.  273-276. 

Approves  of  Drobisch's  solution  of  the  problem,  (himself  having 
appealed  to  Drobisch  for  the  solution  of  it). 

H.  Paasche 

33.  Ueber  die  Preisentwickelung  der  letzten  .Jahre,  nach  den 
Hamburger  Borseunotirungen.  (Jahrbiicher  fiir  Xat.-oekon. 
und  Statistik,  1874,  BandXXIIL,  pp.  168-178.) 

34.  Studien  iiber  die  Natur  der  Geldentwerthung  und  ihre  prak- 
tische  Bedeutung  in  den  letzten  Jahrzehnten. — Jena  1878,  173 
pp.  (Conrad's  Sammlung  uationaloekonomischer  u^d  statis- 
tischer  Abhaudlungen,  Band  I.) 

Follows  Drobisch  in  condemning  both  the  geometric  and  the  arith- 
metic averages,  but,  unlike  Drobisch,  wants  the  mass-quantities  to 
be  considered  only  of  one  period,  preferably  the  later,  [thus  falling 
back  on  Scrope's  method,  like  Laspeyres  himself,  except  for  the  dif- 
ference in  regard  to  the  period  at  which  the  mass-quantities  are 
chosen.  See  Apjjendix  C,  IV.,  |  2  (2)].  Employs  this  method,  in 
the  first  work,  on  twenty  tAvo  articles  on  the  basis  1847-67  through 
the  years  1868  to  1872,  also  with  the  average  of  these  later  years. 
[The  table  is  continued  by  v.  d.  Borght,  below  No.  5.5,  and  Conrad, 
Nos.  96  and  97.] 

A.  Hanauer 

35.  Etudes  economiques  sur  I'Alsace  ancienne  et  moderne.  Vol. 
II.,  Denrees  et  salaires. — Paris  1878.  (Chapt.  XV.,  "  Conclu- 
sion," giving  a  "  resume  general  "  of  the  "  pouvoir  de  Targent 
en  general,"  pp.  601-609.) 

Employs  unevenly  weighted  arithmetic  average  of  the  mass-quan- 
tities of  ten  articles  purchasable  with  one  franc  (5  grammes  silver  at 
9/10)  at  quarter-century  periods  from  1351-75  down,  compared  with 


560  BIBLIOGRAPHY 

the  mass-quantities  purchasable  1851-75  as  units.      [Tlius  virtually 
uses  the  unevenly  weighted  harmonic  average  of  price  variations.] 

A.  EUis 

36.  The  money  value  of  food  and  raw  materials.  (The  Statist, 
London  June  8th  1878.) 

Applies  [Young's]  method  to  twenty  five  articles  for  the  j'ears 
1859,  1869,  1873,  1876  and  the  first  quarter  of  1878,  the  weighting 
being  according  to  the  importance  of  the  articles  in  1869,  which  is 
used  as  the  base. 

Anonymous 

37.  The  rise  in  the  value  of  gold.  (Bankers'  Magazine,  London 
Oct.  1878,  pp.  842-848.) 

Uses  [Dutot's]  method,  comparing  the  prices  of  thirty  one  arti- 
cles in  1878  with  their  prices  in  1868. 

R.  Giffen 

38.  Report  to  the  Secretary  of  the  Board  of  Trade  on  the  prices 
of  exports  of  British  and  Irish  produce  in  the  years  1861-1877. 
(Parliamentary  documents,  Session  1879,  c.  2247,  folio,  pp. 
2-15.) 

39.  On  the  fall  of  prices  of  commodities  in  recent  years.  (Journal 
of  the  Statistical  Society  of  London,  March  1879  ;  republished 
in  Essays  in  Finance,  London  1880,  §  I,  the  extent  of  the  fall, 
pp.  312-322.) 

40.  Report  to  the  Secretarj^  of  the  Board  of  Trade  on  the  prices  of 
exports  of  British  and  Irish  produce,  and  the  prices  of  imports, 
in  the  years  1861-78.  (Parliamentary  documents.  Session 
1880,  c.  2484,  pp.  3-27.) 

41.  Report  to  the  Secretary  of  the  Board  of  Trade  on  recent 
changes  in  the  amount  of  the  foreign  trade  of  the  United 
Kingdom  and  the  prices  of  imports  and  exports.  (Parliamen- 
tary documents.  Session  1881,  c.  3079,  pp.  4-30.) 

42.  Report  to  the  Secretary  of  the  Board  of  Trade  on  recent 
changes  in  the  amount  of  the  foreign  trade  of  the  United 
Kingdom  and  the  prices  of  exports  and  imports.  (Parliamen- 
tary documents.  Session  1885,  c.  4456,  pp.  iii-viii,  1-53.) 

43.  Trade  depression  and  low  prices.  (Contemporary  Review, 
London  June  1885  ;  republished  in  Essays  in  Finance,  Second 
Series,  London  1886,  glll.,  "  The  history  of  prices,"  pp.  16-22.) 

44.  Index  numbers.  (Bulletin  de  I'lnstitut  international  de  Sta- 
tisque,  Rome  1887,  4to.  pp.  126-131.) 

45.  Pvvidence  before  the  Royal  Commission  on  Gold  and  Silver, 
First  Report— London  1887,  qq.  663-793,  folio,  pp.  34-42. 


BIBLIOGRAPHY  561 

Applies  [Young's]  inetliod  to  custom-lioiise  returns,  with  the 
peculiarity  that  while  his  basis  for  prices  in  1861,  he  takes  the 
relative  importance  (according  to  total  money-values)  of  the  vari- 
ous classes  in  the  year  1875  for  the  weighting  through  all  the 
years,  whicli  are  each  compared  directly  with  1801.  [See  Appen- 
dix C,  III.,  ^3.]  Tables  at  first,  in  No.  38,  confined  to  e.xports 
for  the  alternate  years  from  1861  to  1877  ;  later,  in  Xo.  40,  applied 
also  to  imports,  and  extended  to  the  succeeding  years,  and  finally,  in 
No.  42,  carried  backwards  to  1840  for  exports  and  to  1854  for  im- 
ports.    The  primary  object  is  to  measure  tiie  volume  of  foreign  trade. 

S.  Bourne 

46.  On  some  phases  of  the  silver-question.  (Journal  of  the  Sta- 
tistical Society  of  London,  June  1879  ;  on  the  fall  of  prices, 
pp.  413-417.) 

47.  On  the  use  of  index  numbers  in  the  investigation  of  trade  sta- 
tistics. (British  Association  for  the  Advancement  of  Science, 
55th  Meeting,  1885  ;  published  in  the  Report,  London  1886, 
pp.  859-873.) 

48.  Index  numbers  as  illustrating  the  progressive  exports  of  Brit- 
ish produce  and  manufactui-es.  {Ibid.  58th  Meeting,  1888  ;  in 
Report,  1889,  pp.  536-540.) 

49.  Index  numbers  as  applied  to  the  statistics  of  imports  and  ex- 
ports. {Ibid.  59th  Meeting,  1889  ;  in  the  Report,  1890,  pp. 
696-701.) 

In  the  first  employs  [Carli's]  method,  with  strictly  even  weighting, 
but  with  attempt  to  improve  upon  the  table  of  the  Economist  by 
omitting  some  of  the  variations  of  cotton  and  adding  coal,  in  a  table 
from  1847  to  1879.  In  the  others  the  object  is  rather  to  measure  the 
volume  of  foreign  trade,  and  the  index  numbers  are  of  two  kinds, 
referring  to  prices  and  referring  to  volumes.  The  former  employ 
[Scrope's]  method  on  the  mass-quantities  of  1883;  the  latter  com- 
pare the  mass-quantities  which  could  have  been  bought  for  the  sums 
actually  spent  at  the  prices  of  1883 — both  on  sixty  five  articles  in  the 
custom-house  returns  for  several  years  from  1865  down,  [the  latter 
sometimes  involving  the  harmonic  average  of  price  variations]. 

A.  de  Foville 

50.  La  mouvement  des  prix  dans  le  commerce  exterieur  de  la 
France.  (L'  Economistc  frangais,  1st  article,  July  5th  1879, 
folio,  pp.  3-5  ;  2d,  July  19th  1879,  pp.  64-65  ;  3d,  Nov.  1st 
1879,  pp.  533-534.  Second  series,  1st  article,  April  29th  1882, 
pp.  503-505,  2d,  June  17th  1882,  p.  504.) 

51.  Article  "  Prix  "  in  the  Nouveau  Dictionaire  d'Economie 
politique.  Paris  1892,  Vol.  II.  (On  price-measurement,  pp. 
607-612.) 

36 


562  BI]iLIO(;HAPHY 

In  the  first  applies  [James's]  method  [as  already  done  by  Levas- 
seur]  to  the  French  "official  "  and  "actual  values"  so  long  as  pos- 
sible, from  1847  to  1862,  and  thereafter,  down  to  1880,  a  method  by 
comparing  the  "provisional  values"  of  each  year  (on  the  prices  of 
the  preceding  year)  with  its  "actual  values."  ^  [See  in  Appendix 
C,  IV.  (2).  The  object  is  rather  to  measure  the  volume  of  foreign 
trade].     The  second  is  merely  descriptive. 

A.  Messedaglia 

62.  II  cak'olo  dei  valori  medii  e  le  sue  applicazioni  statistiche. 
(Archivio  di  Statistiea,  Auno  V.,  1880.  Republished  by  itself, 
Eome  1883,  86  pp.;  on  prices,  pp.  36-40.) 

Examines  the  matliematics  of  the  tliree  means,  and,  pointing  out 
the  inverse  relationship  between  the  arithmetic  and  tlie  harmonic, 
tliinks  tliat  for  measuring  the  purchasing  power  of  money  over  goods 
we  want  the  arithmetic  average  of  the  variations  of  the  mass-quanti- 
ties, and  therefore  the  harmonic  average  of  the  price  variations,  or 
for  measuring  the  power  of  goods  over  money,  directly  the  arithmetic 
average  of  the  price  variations  ;  but  for  the  geometric  average  there 
is  no  place.     Pays  little  attention  to  weighting. 

H.  C.  Burchard 

53.  In  the  Finauce  Reports  of  the  Secretary  of  the  Treasury, 
Washington,  for  the  years  1881  pp.  312-321,  1882  pp.  252-254, 
1883  pp.  316-318  ;  and  in  the  Report  of  the  Director  of  the 
Mint  on  the  production  of  the  precious  metals  in  the  United 
States,  for  the  year  1884,  pp.  497-502. 

Uses  [Carli's]  method.  The  tables  give  average  prices  for  1883 
and  1884  compared  with  1870,  and  with  1882,  and  also  with  the 
general  average  for  fifty  six  years  ending  1880. 

G.  Fauveau 

54.  Comparaison  du  pouvoir  de  la  monuaie  a  deux  6poques  dif- 
fereuts.  (Journal  des  Economistes,  Paris  June  1881,  pp.  354- 
359.) 

Formulates  [Scrope's]  method. 

R.  V.  d.  Borght 

55.  Die  Preisentwickelung  wiihrend  der  letzten  Decenuien  nach 
der  Hamburger  Borsennotirungen.  (Jahrbiicher  fiir  Nat.- 
oekon.  und  Statistik,  1882,  N.  F.  Band  V.,  pp.  177-185.) 

Continues  Paasche's  tables  [see  above  No.  33]  down  to  1880,  using 
the  same  method. 

H.  Sidgwick 

50.   Principles  of  political   economy. — London   1883.     (Book  I.) 

*  This  mctliod  lias  been  continued  in  several  of  the  Bulletins  du  Minist&re  des 
Finances. 


BIBLIOGRAPHY  563 

Chapt.  II.,  "  On  the  definition  and  measure  of  value,"  pp.  52- 
*       69.) 

Kecommends  [Scrope's]  method,  witli  the  mass-quantities  of  the 
first  or  of  the  second  period,  or  of  a  mean  [the  arithmetic]  between 
tlie  two.  None  of  tliese  being  better  than  another,  there  is  no 
single  authoritative  measurement. 

F.  Y.  Edgeworth 

57.  On  the  method  of  ascertaining  a  change  in  the  value  of  gold. 
(Journal  of  the  Statistical  Society  of  London,  Dec.  1883,  Vol. 
XLVI.,  pp.  714-718.) 

58.  The  choice  of  means.  (The  London,  Edinburgh  and  Dublin 
Philosophical  Magazine  and  Journal  of  Science,  Sept.  1887,  pp. 
268-271.) 

59.  Memorandum  of  the  Secretary,  attached  to  the  First  Report 
of  the  Committee  of  the  British  Association  [see  below  No. 
99],  at  the  57th  Meeting,  1887.  (Published  in  the  Report  of 
that  Meeting,  London  1888,  pp.  254-301.) 

60.  Memorandum  on  the  accuracy  of  the  proposed  calculations  of 
index  numbers,  attached  to  the  Second  Report  of  the  same 
Committee  [see  below  No.  100],  at  the  58th  Meeting,  1888. 
(Published  in  the  Report,  1889,  pp.  188-219.) 

61.  Some  new  methods  of  measuring  valuations  in  general  prices. 
(Journal  of  the  Royal  Statistical  Society,  London,  June  1888, 
pp.  34G-368.) 

62.  Appreciation  of  gold.  (Quarterly  Journal  of  Economics,  Bos- 
ton Jan.  1889,  pp.  151-169.) 

68.  Memorandum,  attached  to  the  Third  Report,  of  the  same 
Committee  as  above  [see  below  No.  101],  at  the  59th  Meeting, 
1889.     (Published  in  the  Report,  1890,  pp.  133-164.) 

64.  Recent  writings  on  index  numbers.  (Economic  Journal,  Lon- 
don 1894,  pp.  158-165.) 

65.  Articles  "Average"  and  "Index  Numbers"  in  Palgrave's 
Dictionary  of  Political  Economy,  Vol.  I.,  London  1894,  p.  74, 
Vol.  II.,  1896,  pp.  384-387. 

66.  A  defence  of  index  numbers.  (Economic  Journal,  March 
1896,  pp.,  132-142.) 

Treats  of  the  measurement  of  the  variations  of  the  value  of  monev 
under  many  [often  artificial]  conceptions  of  the  qucesitum.  Recom- 
mends generally  the  arithmetic  average  with  even  or  uneven  weight- 
ing, but  also  the  median,  and  again  the  geometric  (this  principally 
with  even  weighting),  according  to  the  object  sought. 

L.  Hansard 

67.  On  the  prices  of  some  commodities  during  the  decade  1874-83. 


564  lUHLIOGRAPHY 

(Paper  read  before  the  Bankers'  Institute,  London  Dec.  17th 
1884,  and  published  in  their  Journal,  Jan.  1885,  pp.  1-42.)     ' 
Employs  only  addition  of  the  prices  reduced  on  a  common  scale 

[like  the  Economist  at  first,  virtually  Carli's  method]. 

J.  Lehr 

68.  Beitrage  zur  Statistik  der  Preise,  insbesondere  des  Geldes  und 
des  Holzes. — Frankfurt  a.  M.  1885.  (On  the  method,  pj).  11, 
37-42.) 

Follow' s  Drobisch  in  employ in,<j  double  weighting,  but  seeks  a 
common  unit  for  both  the  periods  compared,  which  he  calls  a 
"pleasure-unit,"  and  which  he  finds  in  the  mass-quantity  of  every 
class  whose  unevenly  weighted  arithmetic  average  price  over  both 
the  periods  is  one  money-unit.     [See  Appendix  C,  V.  ^  2.  ] 

L.  Walras 

69.  D'une  methode  de  regularisation  de  la  valeur  de  la  monnaie. 
(Memoire  read  before  the  Societe  vaudoise  des  Sciences  natur- 
elles,  June  6th  1885;  published,  Lausanne  1885,  22  pp.:  also 
republished  below  in  No.  71.) 

70.  Elements  d'economie  politique  pure. — 2d  ed.,  Lausanne  1889. 
(Pp.  431-432,  457-468,  jiartly  incorporating  the  preceding.) 

71.  Etudes  d'economie  politique  appliquee. — Lausanne  and  Paris 
1898.     (Mostly  reprints,  with  some  new  matter.) 

Follows  Jevons  in  adopting  the  geometric  average  of  price  varia- 
tions with  even  weighting,  and  compares  it  with  the  arithmetic  and 
harmonic  averages,  also  only  with  even  weighting,  all  which  he  for- 
mulates. Later  inclines  to  prefer  what  he  calls  "the  formula  of  the 
multiple  standard"  [Scrope's  method],  because  it  takes  into  account 
the  mass-(]uantities  [of  which  period,  he  does  not  consider]. 

A.  Simon  and  L.  Walras 

72.  Contribution  ^  I'etude  des  variations  des  prix  depuis  la  sus- 
pension de  la  frappe  des  ecus  d'argent.  (Memoire  read  before 
the  Societe  vaudoise  des  Sciences  naturelles,  June  3d  1885, 
published  jointly  with  No.  69,  Lausanne  1885,  11  pp.;  also  re- 
published in  No.  71.) 

Apply  the  geometric  average  with  even  weighting  to  tlie  prices 
of  twenty  articles  at  Berne  from  1871  to  1884  on  the  basis  of  the 
arithmetic  average  of  prices  during  the  period  1871-78. 

[C.  D.  Wright] 

73.  Sixth  annual  Report  of  the  Bureau  of  Statistics  of  Labdr. — 
Boston  18S5,  pp.  154-156. 

Compares  prices  in  Great  Britain  and  in  Massachusetts,  for  work- 
ing men,  using  [Scro])e's]  method  twice  applied,  on  tiie  mass-quan- 
tities of  each  country. 


I!11'.i.1(»(;i;ai'1iv  56o 

M.  G.  Mulhall 

74.  On  the  variations  of  i)rice-level.s  since  1850.  (British  Assoc- 
iation, 55th  Meeting,  1885,  epitomized  in  the  Report,  1886,  pp. 
1157-1158.) 

75.  History  or])rices  since  the  year  1850. — London  1885,  190  pp. 

Emjiloys  [Scrope's]  niethod,  with  the  mass-<iuantities  of  the  later 
periods  [in  Paasche's  form].  Claims  to  apply  it  to  the  whole 
world.     Calls  it  "the  voliinie  of  trade  method." 

S.  Newcomb 

76.  Principles  of  political  economy. — New  York  188G.  (Book 
III.  Chapt.  II.,  "The  measure  of  value  hy  an  absolute  stan- 
dard," pp.  205-214.) 

Recommends  [Scrope's]  method. 

R.  H.  Inglis  Palgrave 

77.  Currency  and  standard  of  value  in  England,  France,  and  India, 
and  the  rates  of  exchange  between  these  countries.  (Memor- 
andum laid  before  the  Royal  Commission  on  Depression  of 
Trade  and  Industry,  1886,  Third  Report,  Appendix  B,  folio, 
pp.  312-390.) 

Gives  various  tables  of  index -numbers,  correcting  the  Econo- 
mist's figures  for  England  by  weighting  the  price  variations  accord- 
ing to  the  money-values  at  the  later  periods  [Young's  method,  more 
specified  as  to  the  weighting,  see  Appendix  C,  III.  §  4],  also  for 
India  and  France,  mostly  from  1865  to  1886. 

F.  B.  Forbes 

78.  The  causes  of  depi-essiou  in  the  cotton  industry  of  the  United 
Kingdom.  (Occasional  Paper  of  the  Bimetallic  League,  No. 
3.)     London  August  1886,  pp.  12,  18,  20. 

Applies  Jevons's  simple  geometric  mean  to  Bai-bour's  figures  of 
quantities  per  rupee,  comparing  1884-85  with  1875-76  on  twelve 
classes  of  exported,  and  seven  of  imported,  goods. 

A.  Sauerbeck 

79.  Prices  of  commodities  and  the  precious  metals.  (Journal  of 
the  Statistical  Society  of  London,  Sept.  1886,  pp.  581-631, 
Appendix,  pp.  632-648.) 

80.  Prices  of  commodities  in  1888  and  1889.     {Ibid.  March  1890, 
.    pp.  141-135.) 

81.  Prices  of  commodities  in  1890.  {Ibid.  March  1891,  pp.  128- 
137.) 

82.  Prices  of  commodities  during  the  last  seven  years.  {Ibid. 
June  1893,  pp.  215-238,  Appendix,  pp.  239-247  and  254.) 

83.  Prices  of  commodities  in  1893.  {Ibid.  March  1894,  pp.  172- 
183.) 


566  BIBLIOGRAPHY 

84.  Prices  of  commodities  in  1894.  (Ibid.  March  1895,  pp.  140- 
154.) 

85.  Index  numbers  of  prices.  (Economic  Journal,  June  1895,  jtp. 
161-174.) 

86.  Prices  of  commodities  in  1895.  (Journal  of  the  Statistical  So- 
ciety, March  1896,  pp.  186-201.) 

87.  Prices  of  commodities  in  1896.  (Ibid.  March  1897,  pp.  180- 
194.) 

88.  Prices  of  commodities  in  1897.  (Ibid.  March  1898,  pp.  149- 
162.) 

89.  Prices  of  commodities  in  1898.  {Ibid.  March  1899,  pp.  179- 
193.) 

90.  Prices  of  commodities  in  1899.  {Ibid.  March  1900,  pp.  92- 
106.) 

Applies  [Carli's]  method  to  the  prices  of  forty  five  articles  on  the 
bases  of  prices  in  1867-77,  going  back  to  1848  and  continuing  to  the 
present.  Also  adds  (beginning  in  No.  82)  two  "tests,"  which  seem 
to  be  [Young's]  method,  on  the  relative  money-values  in  1889-91, 
and  [Scrope's]  method  on  the  mass-quantities  of  the  other  years 
[like  Paasche's].  In  No.  86  (pp.  193-194)  he  experiments  with  the 
geometric  mean. 

F.  Coggeshall 

91.  The  arithmetic,  geometric,  and  harmonic  means.  (Quarterly 
Journal  of  Economics,  Oct.  1886,  pp.  83-86.) 

Discusses  the  three  means,  mostly  from  the  point  of  view  of  avoid- 
ing error  in  the  result  arising  from  errors  in  the  data,  witliout  attach- 
ing superiority  to  any  one  of  them,  regarding  the  mean  of  prices  as  a 
"fictitious  mean." 

J.  E.  Thorold  Kogers 

92.  A  history  ul' agriculture  and  prices  in  England,  Vol.  V. — Ox- 
ford 1887.  (Chapt.  XXVI.,  "On  prices  generally  between 
1583  and  1702,"  pp.  778-800.) 

Employs  [Carli's]  method. 

A.  Marshall 

93.  Remedies  for  fluctuations  of  prices.  (Contemporary  Review, 
London  March  1887,  Sect.  V.,  "  How  to  estimate  aunitof  pur- 
chasing power,"  pp.  371-375.) 

Assumes  tlie  arithmetic  average. 
J.  S.  Nicholson 

94.  The  measurement  of  variations  in  the  value  of  the  monetary 
standard.  (Paper  read  before  the  Royal  Society  of  Edinburgh, 
March  21st  1887;  published  in  the  Journal  of  the  Statistical 


BIBI.IOGRAPJIY  5<)7 

Society  of  London,  March  1887,  and  republished  in  Treatise  on 
money,  Edinburgh  and  London  1888,  pp.  298-331.) 

Invents  a  new  method  [which  is  partly  a  variation  upon  Drobisch's. 
See  Appendix  C,  V.  g  3]. 

A.  Beaujon 

95.  Sur  la  question  des  "  index  numbers." — Propositions  soumises 
k  I'Institut  international  de  Statistique  en  vue  d'obtenir  des 
tableaux  de  prix  moyens  comme  base  du  calcul  des  index  num- 
bers.— Index  numbers  ou  chiftres  de  prix  de  marchandisesdans 
divers  etats,  depuis  1870.  (Bulletin  de  I'Institut  international 
de  Statistique,  1887,  pp.  106-114,  115-116,  117-126.) 

Discusses  manner  of  collecting  data.  Leaves  the  question  of  aver- 
ages to  a  future  deliberation  of  the  Institute.^  In  the  third  reports 
several  tables  of  the  writers  above. 

J.  Conrad 

96.  Beitriige  zur  Beurteilung  der  Preisreduktion  in  den  80  er 
Jahren.  (Jabrbiicher  fiir  Kat.-oekon.  und  Stati&tik,  1887,  N. 
F.  Band  XV.,  pp.  322-331.) 

97.  Die  Entwickelung  des  Preisniveaus  in  den  letzten  Decennien 
und  der  deutsche  Getreidebedarf  in  den  letzten  Jahren.  (Ibid. 
1899,  DritteF.  Band  XVII.,  i)p.  642-660.) 

Continues  Paasche's  and  v.  d.  Borght's  tables  down  to  1885,  and 
later  to  1897,  using  the  same  method. 

F.  Krai 

98.  Geldwertund  Preisbewegung  im  Deutschen  Reiche  1871-1884. 
— Jena  1887.     (On  prices,  pp.  63-111.) 

Uses  the  arithmetic  average  with  both  [Dutot's]  haphazard  weight- 
ing and  [Evelyn's]  weighting  by  classification. 

British  Association  for  the  Advancement  of  Science : 

Committee  consisting  of  S.  Hourne,  F.  Y.  Edgowortb,  H.  S.  Fox- 
well,  R.  Giffen,  A.  Marshall,  J.  B.  Martin,  J.  S.  Nicholson,  R. 
H.  I.  Palgrave  and  H.  Sidgwick,  appointed  for  the  purpose  of 
investigating  the  best  methods  of  ascertaining  and  mea.suring 
variations  in  the  value  of  the  monetary  standard. 

99.  First  report,  to  the  57th  Meeting,  1887.  (Published  in  the 
Report,  London  1888,  pp.  247-254.) 

*A  Comite  de  la  Statistique  des  Prix  was  appointed  by  the  Institute,  consisting 
of  Beaujon,  de  Foville,  de  luama-Sternegg,  Gilt'en,  de  Neuraann-Spallart,  de  Mayr, 
and  Paiitaleoni.  But  beside  brief  reports  on  recent  works,  made  by  Martin  and 
Palgrave,  October  1891  and  September  1893  (in  the  Bulletin,  1892,  pp.  24^-246, 
and  189."),  pp.  57-60),  this  Committee  does  not  appear  to  have  made  an  original 
report. 


568  BIBLIOGRAPHY 

100.  Second  report,  to  the  58th  Meeting,  1888.  {Ibid.  1889,  pp. 
181-188.) 

101.  Third  i-eport,  to  tlie  5!)th  Meeting,  1889.     {Ibid.  1890,  p.  133.) 

102.  Fourth  report,  to  the  60th  Meeting,  1890.  {Ibid.  1891,  pp. 
485-488.) 

Review  several  forms  of    [Scrope's]  method,  and  discuss  various 
(juestions  eonnected  with  ])rioe-measurements. 

E.  Nasse 

103.  Das  Sinken  der  Warenpreise  wiihreud  der  letzten  fiinfzehn 
Jahre.  (Jahrbiicher  fur  Nat.-oekon.  und  Statistik,  1888,  N.  F. 
Band  XVII.;  on  methods  of  measurement,  pp.  51-53.) 

104.  Das  Gehl  und  Miinzwesen.  (Schonbergs  Handbuch  der  Po- 
litischen  Oekonomie,  Tiibingen,  Vol.  II.,  1890;  on  methods  of 
measurement,  pp.  331-332,  folio.) 

Reviews  some  of  the  methods  using  the  arithmetic  average. 

K.  Wasserab 

105.  Preise  und  Krisen.  Gekronte  Preisschrift  iiber  die  Veran- 
derungen  der  Preise  auf  dem  allgemeinen  Markt  seit  1875  und 
deren  Ursachen. — Stuttgart  1889,  (On  price-measurements, 
pp.  75-128.) 

Applies  a  method  of  liis  own   [whicli  is  really  Young's]  to  prices 
on  the  basis  1861-70  down  to  1885. 

F.  Schmid 

lOG.  Berioht  iiber  die  Thatigkeit  des  statistischeu  Seminars  an  der 
k.  k.  UniversitJit  Wien  im  Wintersemester  1888-89.  (Statis- 
tischeMonatschrift,  XY  Jahrgang,  Vienna  1889.  On  variations 
in  the  i)urchasing  power  of  money,  aided  by  G.  H.  Thierl,  pp. 
648-650  ) 

Reviews  several  methods. 

E.  B.  Andrews 

107.  An  honest  dollar.  (Publications  of  the  American  Economic 
Association,  New  York,  Vol.  IV.,  No.  6,  Nov.  1889  ;  see  pp. 
38-39.) 

108.  Institutes  of  economics. — Boston  1891,  pp.  141-142. 

lieconuiionds  [Scrope's]    method,  hut  allows  the  use  of  the  geo- 
metric, aritlnnetic,  or  liarmonic  means. 

K.  T.  von  Ihama-Sternegg 

109.  Der  Kiickgang  der  Waarenpreise  und  die  oesterreichisch- 
ungarische  Handelsbilanz  1875-1888.  (Statistische  Monat- 
schrift,  XVI  Jahrgang,  1890.     Tables,  pp.  6-7.) 


i!ii'.M«)(;i!Ai'in'  509 

(lives  tables  aj)i)lying  [Carli's]  rnetliod  (in  a  lorm  like  iIk;  Kcono- 
inist's)  to  thirty  articles  of  import  and  to  twenty-five  of  export  in 
Austria-Hungary  from  1880  to  1888  on  tlie  basis  of  1875-1879. 

H.  Westergaard 

110.  Die  Grundziige  der  Theorie  der  Statistik. — Jena  1<S90.  (On 
price-measurements,  pp.  218-220.) 

Points  out  that  the  geometric  average  [lie  supj)()ses  the  same 
weighting  throughout,  having  even  weighting  mostly  in  mind]  gives 
the  same  index-numbers  in  a  series  of  periods  whether  applied  to 
comparing  each  subsequent  period  with  tlie  original  base  or  to  com- 
paring any  of  the  subsequent  periods  with  each  other ;  and  that  this 
is  not  done  by  the  usual  methods  employing  the  aritlimetic  average 
[Carli's  and  Young's].  <  )Hers  this  as  an  argument  for  the  geometric 
average. 

E.  P.  Falkner 

111.  Report  of  the  Statistician.  (Pp.  xi-c  in  Vol.  I.  of  the  Re- 
port on  retail  prices  of  Mr.  Aldrich  from  the  Committee  on 
Finance,  52d  Congress,  1st  Session,  No.  986.  Washington 
1892.) 

112.  Report  of  the  Statistician.  (Pp.  27-337  in  Vol.  I.  of  the  Re- 
port on  wholesale  prices  of  Mr.  Aldrich  from  the  Committee 
on  Finance,  52d  Congress,  2d  Session,  No.  1394.  Washington 
1893.) 

113.  Wholesale  prices:  1890  to  1899.  (Bulletin  of  the  Depart- 
ment of  Labor,  No.  27,  Washington  March  1900  ;  pp.  237-313.) 

Applies  both  [Carli's]  and  [Young's]  methods  to  prices  in  the 
I'nited  States,  in  the  first  comparing  1891  with  1889,  in  the  second 
extending  the  investigation  to  the  years  1840-1891  on  the  basis  of 
18<)0.  and  in  the  third  bringing  it  down  to  1899. 

S.  McC.  Lindsay 

114.  Die  Berechnung  der  Edelmetalle  seit  1850. —Jena  1893. 
(Conrad's  Sammlung.     On  the  method,  pp.  9-28.) 

Reviews  several  methods,  and  adopts  [Scrope's]  with  the  mass- 
quantities  of  the  first  or  of  the  second  period,  or  a  mean  between 
them,  according  lis  any  of  tliese  best  represents  the  importance  of 
the  classes. 

R.  Zuckerkandl 

115.  Die  statistische  Bestimmung  des  Preisniveaus.  (Handwor- 
terbuchder  Staatswissenschaften  Jena,  Vol.  V.,  1893,  4to.,  pp. 
242-251.) 

116.  La  mesure  des  transformations  de  la  valeur  de  la  monnaie. 
(Revue  d'economie  politique,  Paris  1894,  pp.  237-253.) 


570  BIBLIOGRAPHY 

Says  tliat  [Scrope's]  method  is  correct  when  the  mass-quantities 
are  tlie  same  at  both  periods;  wlien  they  are  different,  Lehr's  method 
comes  the  nearest  to  the  truth. — For  the  purpose  of  paying  contracts 
with  money  of  stable  purchasing  power,  recommends  another  method, 
which  is  not  clearly  described,  [but  which  seems  to  revert  to  Scrope's, 
applied  to  the  mass-quantities  at  the  time  of  making  the  contract]. 
(The  second  is  a  translation  of  the  first,  with  a  few  changes.  Each 
contains  a   brief  bibliography. ) 

G.  d'Avenel 

1 1 7.  Histoire  economique  de  la  in-opriete,  des  salaires,  des  denr6es, 
et  de  tons  les  prix  en  general  depuis  Tan  1200  jusqn'eu  I'an 
1800.— Paris,  4  vols.,  4to.,  1894-1898.  (On  the  method,  Vol. 
I.,  pp.  6-13  ;  results,  pp.  27,  32,  137.) 

118.  La  fortnue  privee  a  travers  sept  siecles. — Paris  1895,  16mo. 
(On  the  methods,  pp.  4-14  ;  results,  p.  37.) 

Apparently  draws,  very  roughly,  an  average  (the  arithmetic)  of 
the  mass-quantities  of  goods  purchasable  with  given  amounts  of  silver 
at  different  epochs  from  1200  to  1890,  [thus  using  the  harmonic 
average  of  their  prices],  with  uneven  weighting  (in  budgets  of  the 
expenditures  of  three  different  classes  of  society).  (The  second  an 
abridgement  of  the  first.) 

T.  H.  Whitehead 

119.  The  critical  position  of  British  trade  with  Oriental  countries. 
(Paper  read  before  the  Royal  Colonial  Institute,  Feb,  12th  1895, 
and  reprinted  from  the  Proceedings  of  the  Institute.  Table  on 
p.  35.) 

Gives  index-numbers  compiled  by  W.  S.  Wetmore,  on  the  plan  of 
the  Economist  [Carli's  method],  for  twenty  articles  in  China. 

J.  A.  Fraser  and  C.  H.  Sergei 

120.  Sound  money. — Chicago  1895.     (A  measurement,  p.  171.) 

Casually  apply  [Dutot's]  method  to  four  articles,  comparing  1871 
and  1891. 

F.  W.  Taussig 

121.  Results  of  recent  investigations  on  prices  in  the  United  States. 
(Bulletin  de  I'lnstitut  international  de  Statisque,  1895,  pp.  22- 
32.) 

Review.s  Falkner's  works  (Nos.  Ill  and  112)  and  discusses  some 
points  in  connection  with  index-numbers. 

N.  G.  Pierson 

122.  IikIcx  numbers  and  the  appreciation  of  gold.  (Economic 
Journal,  September  1895,  pp.  329-335.) 


HTIM.IOOKAl'HV  571 

Is  content  with  tlie  aritlimotic  average.  Prefers  Soetbeer's  results 
because  of  the  great  number  of  articles  used.^ 

A.  L.  Bowley 

123.  Comparison  of  the  rates  of  increase  of  wages  in  the  United 
States  and  in  Great  Britain  1860-1891.  (Paper  read  before  the 
British  Association,  Sept.  12th  1895  ;  published  in  the  Economic 
Journal,  1895  ;  price-measurements,  p,  381.) 

Uses  the  arithmetic  average. 

a.  Wiebe 

124.  Zur  Geschichte  der  Preisre volution  des  XVI  und  XVII  Jahr- 
hunderts. — Leipzig  1895.  (On  the  method,  pp.  163-174 ; 
tables,  pp.  369-38G.) 

Applies  both  [Carli's]  and  [Evelyn's]  methods  to  European  prices 
from  14.51  to  1700.  Recommends  Lehr's  method  for  present  re- 
searches. 

H.  Denis 

125.  La  depression  economique  et  sociale  et  I'histoire  des  prix. — 
Ixelles-Bruxelles  1895.     (On  price-measurements,  pp.  9-34.) 

Uses  the  arithmetic  average  with  even  weighting  [Carli's  method]. 
This  weighting  he  considers  sufficiently  approximative  in  practice, 
though  wrong  in  theory. 

G.  M.  Boissevain 

126.  La  question  monetaire.  (Memoire  traduit  du  Hollandais.) 
— Paris  1895.     (On  the  method,  pp.  52-53  ;  and  two  tables.) 

Modifies  Sauerbeck's  index-numbers  by  calculations  upon  British 
exports  and  imports. 

A.  L.  Fonda 

127.  Honest  money. — New  York  1895,  12  mo.  (On  "the  stan- 
dard of  value,"  pp.  158-161,  165.) 

For  his  standard  would  use  [Young's]  method,  to  be  applied  to 
a  hundred  staple  commodities. 

Wharton  Barker 

128.  The  course  of  prices.  (The  American,  Jan.  25th  1896,  folio, 
pp.  54-56,  and  quarterly  since.) 

Gives  index-numbers  for  American  prices  on  a  hundred  and  one 
articles  in  thirteen  groups  from  Jan.  1st  1891,  quarterly,  continued 
to  the  present,  using  the  arithmetic  average  of  the  price  variations 
with  even  weighting,  all  compared  with  the  first  period  (following 
the  example  of  the  Economist  in  England). 

*  In  Farther  consideratio7is  on.  indfx-numbers,  in  the  same  Journal,  March 
1896,  pp.  127-1.31,  he  rejects  the  whole  system  of  index-numbers  because  different 
results  can  be  obtained  on  the  same  price  variations  [by  using  different  weights]. 
(Eeply  by  Edgeworth,  No.  66.) 


572  BIBLIOGRAPHY 

J.  Allen  Smith 

129.  The  multiple  money  standard.  (Publications  of  the  Ameri- 
can Academy  of  Political  and  Social  Science.  Philadelphia 
1896.     On  the  measurement  of  the  standard,  pp.  27-30.) 

For  his  standard  would  use  [Scrope's]  method,  the  mass-quantities 
chosen  to  be  revised  from  time  to  time  (at  long  intervals),  all  com- 
modities being  included  that  can  be  accurately  defined  as  to  quantity 
and  ((uality. 

T.  N.  Whitelaw 

130.  A  contribution  to  the  study  of  a  constant  standard  and  just 
measure  of  value. — Glasgow  1896.  (On  the  standard,  pp.  18- 
19,  32-35.) 

For  his  standard  would  use  [Carli's]  method,  to  be  applied  to 
about  twenty  of  the  chief  agricultural  products. 

L.  G.  Powers 

131.  Fifth  annual  report  of  the  Bureau  of  Labor  of  the  State  of 
Minnesota,  1895-1896.— St.  Paul  1896,  524  pp  (On  the 
methods  used,  pp.  26-30.) 

Uses  (1)  [Drobisch's]  method  in  groups  of  articles  whose  prices 
are  reported  in  the  same  mass-unit,  (2)  [Scrope's]  method  applied  to 
the  arithmetic  average  of  the  mass-quantities  over  thirty  five  years, 
(3)  Sauerbeck's  "corrected  method"  [Paasche's,  or  Scrope's  ap- 
plied to  the  mass-quantities  of  the  later  years  singly],  and  (4)  the 
"simple  "  arithmetic  average  of  the  price  variations  [Carli's  method]. 
Calculations  confined  to  agricultural  products,  principally  in  the 
West,  extending  from  1862  to  1895,  on  the  basis  of  1872.  [Are 
vitiated  by  combining  the  prices  of  widely  separated  localities,  and 
by  attempting  to  eliminate  the  efi'ects  of  reduced  costs  of  transporta- 
tion. 1 

M.  Bourguin 

132.  La  raesure  de  la  valeur  et  la  monnaie. — Paris  1896,  273  pp. 
(On  index  numbers,  pp.  134-139.) 

Although  denying  the  existence  of  general  exchange-value,  wants 
to  measure  the  average  of  the  variations  of  all  the  particular  ex- 
change-values of  money.  Approves  of  the  arithmetic  average,  with 
even  weighting. 

L.  L.  Price 

133.  Money  and  its  relations  to  jirices — being  an  enquiry  into  the 
causes,  measurement,  and  eftects  of  changes  in  general  prices. 
—London  1896,  12  mo.     (On  the  measurement,  pp.  9-36.) 

Briefly  surveys  the  subject. 
F.  J.  Atkinson 

134.  Silver  prices  in  India.  (Journal  of  the  Royal  Statistical 
Society,  March  1897,  pp.  84-147.) 


BIBLIOGRAPHY  573 

Applies  [Young's]  method  to  one  hundred  articles  in  forty  groups 
in  various  parts  of  India  from  1801  to  189'),  on  the  price-basis  of 
1871,  with  weighting  according  to  the  total  money-values  of  the 
groups  in  1893-94. 

Eltweed  Pomeroy 

135.  The  multiple  standard  for  money.  (Arena,  Boston  Sept. 
1897.     On  the  method,  pp.  331-338.) 

For  his  standard  would  use  [Scrope's]  method  applied  to  the  mass- 
quantities  consumed  by  working  men's  families,  employing  two  hun- 
dred staple  articles,  their  prices  being  collected  from  one  hundred 
centers  of  commerce. 

F.  Parsons 

136.  Eatioual  money.  A  national  currency  intelligently  regulated 
in  reference  to  the  multiple  standard. — Philadelphia  1898,  (Ou 
the  standard,  pp.  113-138.) 

For  his  standard  would  use  [Scrope's]  method  applied  to  a 
couple  of  luindred  articles  in  the  mass-quantities  that  are  consumed 
by  the  average  family,  the  list  to  be  revised  from  time  to  time 
(frequently). 

R.  Mayo-Smith 

137.  Movements  of  prices.  (Political  Science  Quarterly,  New 
York  Sept.  1898,  pp.  477-494.) 

188.  Statistics  and  economics. — New  York  1899.  (Chapt.  VI., 
"Prices";  on  the  measurement,  pp.  199-228.) 

Briefly  reviews  the  problems  connected  with  index-numbers.  (The 
second  slightly  expanded  from  the  first. ) 

K.  Wicksell 

139.  Geldzins  und  Giiterpreise. — Jena  1898.  (On  price-measure- 
ments, pp.  6-16.) 

Eecommends  [Scrope's]  method,  provided  the  results  are  the 
same  on  the  mass-quantities  of  each  period.  Otherwise  the  problem 
is  unsolvable,  as  the  measurement  with  the  mass-quantities  of  the 
one  period  is  as  good  as  with  those  of  the  other,  and  the  mean  be- 
tween the  two  can  have  only  "conventional  meaning." 

A.  W.  Flux 

140.  Some  old  trade  records  re-examined  :  a  study  in  price-move- 
ments during  the  present  century.  (Paper  read  Feb.  8th 
1899  before  the  Manchester  Statistical  Society,  and  published 
in  their  Tiansactions,  Session  1898-99,  pp.  65-91.) 

Applies  [James's]  method  to  British  jirices  1798-1869,  to  French 
prices  1873-97  [cf.  De  Foville,  No.  50],  and  to  German  prices 
1891-97. 


574  HrBi.i()(;i;APHY 

R.  S.  Padan 

141.   Prices  and  index  numbers.     (Journal  of  Political  Economy, 
Chicago  March  1900,  pp.  171-202.) 

Considers  the  arithmetic  average  the  only  rigorous  one.  Attacks 
Jevons  for  "beclouding"  the  subject  by  introducing  the  geometric 
average  and  suggesting  the  harmonic.  Advocates  recognition  of 
mass-quantity.  This  being  done,  the  method  [Scrope's]  bethinks 
to  be  accurate  provided  the  mass-quantities  are  proportional  at  all 
the  periods  compared,  because  then  the  same  results  are  obtained 
of  whatever  period  the  mass-units  be  used.  But  not  so,  if  the  mass- 
(piantities  are  irregular,  so  that  in  such  cases  no  one  result  is  authori- 
tative. 


INDEX 


[Simple  numernh  refer  to  pacjett.     Following  the  letter  B  tJiey  refer  to  works  in 
the  Bihliocjraphy.'} 


Absolute  :  of  value  3n. ;  of  variations 
57-8,  66  ;  its  meaiiinjjs  66-7. 

Absoluteness,  want  of,  no  defect  in 
mensuration  65-6. 

Aggregate  of  exchange-values,  varia- 
tions of  439,  459  ;  measurement  of 
its  variations  459-61. 

Appearance  of  new  classes  112-14. 

Aldrich  Report  475-6  ;  see  B.  111-12. 

Andrews  480n.,  540  ;  B.  107,  108. 

Approach  :  correction  bv  103-4,  105, 
117,  118,  420-3. 

Aristotle  123n.,  137n. 

Arithmetic  average  and  mean  :  pref- 
erence for  218-19,  222-3  ;  careless 
rejection  of  223,  546n.;  argument 
for,  in  general,  refuted  249-51, 
253-4  ;  argument  for,  in  averaging 
prices  229,  257-8  ;  disproved  when 
the  sums  are  constant  296-8  ;  study 
of  502-6. 

Arrivabene  438n. 

Atkinson,  F.  J.,  B.  134. 

Avenel,  G.  d'  219n.,  477n. ;  B.  117- 
18. 

Average  :  led  to  by  the  formulae  145  ; 
suggested  216  ;  comparison  of,  and 
average  of  variations  157,  159  (cf. 
164),  184,  185,  Appendix  A.;  must 
be  constructed  158-9,  185  ;  ques- 
tion of,  raised  by  Jevons  220,  223  ; 
the  geometric,  to  be  distinguished 
from  the  geometric  mean  239, 
241-3,  255,  436  ;  not  so  with  the 
arithmetic  and  harmonic  239-40  ; 
nature  of,  in  general  498-501. 

Bailey  23n. 

Bankers'  Magazine  534;  B.  37. 

Barbour  219n. 

Barker  536  ;  B.  128. 

Bascom  43Sn. 

Basic  period,  unimportance  of  208. 

Beaujon  80n.,  97n.,  567n.;  B.  95. 

Beccaria  xvi,  3n.,  7n.,  lln. 


Boissevain  B.  126. 

Bo]les37n.,  438n. 

Borght,  R.  V.  d.  99n.,  541  ;  B.  55. 

Bourguin  7n.,  Sn.,  lOn.,  12-13n., 
69n.,  140n.,  438n.;  B.  132. 

Bourne  536,  541,  550,  5()7 ;  B.  46-9, 

Bowley  B.  123. 

British  Association  for  tlie  Advance- 
ment of  Science,  Reports  of  its 
Committee,  85n.,  86n.,  97n.,  113n., 
123n.,  207n.,  495n. ;  B.  99-102. 

Budelius  137n. 

Burchard  536  ;  B.  53. 

Cairnes  27n.,  438n.,  439. 

Cantillon  132n. 

Carli  82,  220  ;  his  metliod  188,  193, 
203,  204,  383,  432,  534-6  ;  B.  4. 

"  Catallactics  "  6n. 

Causes :  not  to  be  sought  for  in 
measurements  22-5,  26-7,  32,  483- 
4 ;  an  instance  where  wrongly 
used  37-8. 

Cibrario  79n. 

Clark,  J.  B.,  8n. 

Classes  of  things  having  exchange- 
value  77. 

Clement,  A.  439n. 

Coggeshall  B.  91. 

Coincidence :  of  the  averages  or 
means,  when  the  sums  are  con- 
stant 306-7,  318n. ;  when  the  masses 
are  constant  350-5,  510-14,  519-20, 
545n. ;  of  the  three  principal  meth- 
ods, with  two  equally  important 
classes  402-3;  of  still  another  with 
them  413. 

Commodity-standard  465. 

Compensation  47-8,  52-3  ;  requires 
equality  233-4  ;  aritiimetic,  har- 
monic, and  geometric  238-9. 

Condillac  79n.,  137n. 

Conrad  99n.,  479n.,  541  ;    B.  96-7. 

Conservation  of  exchange-value  51, 
438-51. 


575 


57G 


INDEX 


Constancy  of  general  exchange-value 
possible  44-53. 

Consumer,  the  average  85. 

Cost  distinguislied  from  price  124-5n. 

Cost-value  :  described  1-4  ;  measure- 
ment of,  often  confovuided  with 
measurement  of  exchange-value 
24-5  ;  one  thing  alone  may  rise  or 
fall  in  38-9  ;  measured  by  (juan- 
titv  of  labor  required  to  produce 
the  thing  57,  6Gn.,  124,  126  n. ; 
we  want  all  things  to  fall  in  484, 
489. 

Courccllc-Seneuil  7n.,  15n.,  438  n. 

Cournot38n.,  (30,  67,  69. 

Cross,  W.  479n. 

Cunningham  208n. 

Dabos,  II.  lOn. 

Davenport,  II.  J.  12on. 

Del  Mar  482n. 

Denis  10  n.,  438n.;  B.  125. 

Deviation  of  tlie  geometric  average 
241-2,  313-24,  363-8,  517-519,  521. 

Dick,  G.  H.  122n. 

Disappearance  of  old  classes  113-14. 

Drobisch  41  n.,  84n.,  86n.,  97,  98,  99, 
111,  113n.,  182, 194,  225,  387n.,  540, 
541,  542  ;  his  raetliod  194-201,  203, 
204,  205,  211,  261 -2n.,  383-6,  388, 
390,  392,  396,  427,  544,  547,  548, 
550,  551,  559;  B.  29-31. 

Dupr^  de  Saint-Maur  B.  3. 

Dutot  82,  122n.,  220,  276;  his  method 
188,  203,  279n.,  534,  554;  B.  2. 

Earnings  127  ;  see  Wages. 

Economist,  The  :  its  index-numbers 
83  ;  its  method  189,  432,  536,  561, 
565  ;  see  B.  21. 

Eden  85n. 

p]dge\vorth  77n.,  85n.,  86n.,  122n., 
123n.,  179n.,  180n.,  203n.,  207n., 
222,  223n.,  224n.,  537,  538,  540, 
541,  542,  543,  544,  546,  548,  567, 
571 n.;  B.  57-66. 

Ellis  537  ;  B.  560. 

P>llis.sen,  A.  31  n. 

Ely,  K.  T.  43Sm.,  4;;9ii. 

Engel  85n. 

I*x]uality  :  in  opi)osite  variations  29- 
30 ;  in  compensatory  variations, 
three  ways  of  conceiving  it  234ff., 
of  distance  travei-sed  236-7,  238, 
cf.  24(i ;  of  i)rop(.rtion  237,  238  ; 
arithmetic,  harmonic,  and  geo- 
metric 237-8  ;  indefinitely,  the 
geometric  244. 

Error,  measurement  of,  in  the  method 
for  constant  sums  332-41,   in   tlie 


universal  methods  394,  402,  404-7, 
424-7. 

Esteem-value  :  described  1-6  ;  meas- 
urement of,  often  confounded  with 
the  measurement  of  e.xchange-value 
24-5  ;  one  thing  alone  may  per- 
iiapsrise  or  fall  in  38-9  ;  measured 
by  esteem  (and  this  by  the  quantity, 
or  supply,  of  the  thing)  58,  66n., 
also  by  the  (juantity  of  labor  the 
thing  will  command  r26n.;  com- 
l)arcd  with  cost-value  124-5 ; 
measurement  of,  desirable  25,  125  ; 
measurement  of,  of  commodities  in 
general  125,  127-8 ;  confusion  in 
combining  this  measurement  with 
the  measurement  of  the  exchange- 
value  of  money  128-34 ;  we  want 
all  things  to  fall  in  484. 

Evelyn  82,  84n.,  122n.,  132n.,  208n.; 
his  variety  of  Carli's  method  188, 
432,  535,  536,  555,  556,  558  ;  B.  5. 

Exchange-value  :  described  1-6  ;  con- 
founded with  otlier  kinds  of  value 
4-6,  23-4,  38-9,  134,  484;  rela- 
tively of  7-13,  55,  this  not  peculiar 
56ft".;  is  a  power  7,  12  ;  a  property 
q/"and  in  things  7,  9-10  ;  is  not  the 
other  thing  11,  nor  the  mere  rela- 
tion with  it  11;  variability  of  10, 
44-5,  480-1  ;  is  local  and  temporal 
14,  208n.,  480;  involves  measure- 
ment 21-2  ;  measurement  of,  to  be 
kept  distinct  from  tliat  of  other 
kinds  of  value  25,  133;  particular 
10-13,  457  and  n.;  individual  91; 
general  5n.,  12-13n.,13;  two  kinds 
of  general  13,  39-41,  (this  distinc- 
tion not  peculiar  67-70),  their  con- 
sistency 43,  their  different  beiiavior 
30,  31,  41-2,  48-51,  relation  be- 
tween their  variations  454-7,  461-3, 
need  of  care  to  distinguish  them 
468-71  ;  one  thing  alone  cannot 
change  in  36-9;  nor  can  all  things 
rise  or  fall  in,  together  438-9  ;  cf. 
441  ;  emptiness  of  the  desire  that 
all  things  should  fall  in  484-7  ; 
mensuration  of,  like  that  of  other 
simple  attributes  64-6,  with  a  pe- 
culiarity 70-1,  how  to  meet  this 
72-4  ;  wc  may  attain  to  a  method 
of  measuring,  if  not  to  a  measure 
of  65  ;  amount  of  precision  needed 
74-5. 

Falkner  85n.,  98,    438n.,  536,  537  ; 
^  B.  111-13. 
Familv  budgets  85n. 
Fauveaull3n. ;  540;  B.  54. 


INDEX 


577 


Fawcett  4,  37n. 

Ferrara,  F.  45n. 

Fianiingo  25n. 

Fleetwood  B.  1. 

Flux,  A.  W.  542n.;  B.  140. 

Fonda  84n.,  123n.,  143n.,  438n., 
471hi.,  493n.,  537  ;  B.  127. 

Forbes  222n. ;  B.  78. 

Fornuilatioii  :  by  equivalertce  18, 
138,  171-2  ;  by  equality  of  ex- 
change-value 139  ff. ;  by  prices  173 
ft. ;  reciprocal  correspondence  be- 
tween these  175-6  ;  with  uneven 
weighting  150-4,  178-80;  with 
double  weighting  154-6,  181-4, 
521-3,  544-52 ;  using  weighting 
not  intended  159-69,  185-94,  535, 
537,  541,  ignorantly  introducing 
double  weighting  193-4,  538-9, 
541-2;  of  exchange-value  in  all 
things  208-11,  444-6,  457-9. 

Foville,  A.  de  96n.,  203n.,  542,  567n. ; 
B.  50,  51. 

Fox  well  85n.,  207n.,  567. 

Fraser,  J.  A.  534;  B.  120. 

Frost,  O.  J.  479n. 

Galiani  45n.,  137n. 

Garnet,  L.  A.  29n. 

(iarnier  29 n.,  (>5-6n. 

Gauss  409n.,  424n. 

Geometric  average :  behaves  differ- 
ently from  the  geometric  mean,  see 
Deviation  and  Average  ;  studv  of 
514  16. 

<  reometric  mean  :  generally  to  be 
used  in  relations  or  ratios  105,  220; 
should  be  used  in  averaging  weights 
105-10  ;  general  argument  for  229- 
30,  231-2,  233ft:  ;  generally  to  be 
used  in  variations  244,  251-3,  254- 
5  ;  see  also  Coincidence. 

Gever  546n.,  558;  his  method  536; 
B.  28. 

Gide  38n. 

Giffen84n.,96n.,  lOOn.,  122n.,  132n., 
478n.,  554n.;  his  method  537-8, 
567  and  n. ;  B.  38-45. 

Gold  and  Silver  Commission  551. 

Grotius  478  and  n. 

< tunton  475n. 

Hallam  13n. 

Hanauer  219  ;  B.  35. 

Hansard  438n.,  536  ;  B.  67. 

Harris  127n. 

Plarmonic  average  and  mean  :  argu- 
ment for,  in  general  251  ;  argument 
for,  in  averaging  prices  256-7  (cf. 
215,  228-9),  defect  in  this  2()0-2, 

37 


refutation  of  it  when  the  sums  are 
con.stant  294-6  ;  study  of  506-10. 

Ilearn  6n. 

Hegel  353n. 

Heitz  97n. 

Held  45n. 

HeUIerieh,  K.  25n. 

Holt,  B.  W.  128-9n. 

Horton  84n.,  85n.,  473,476,  478n. 

Hyde  122n. 

Inama-Sternegg,  K.  T.  von  567n.;  B. 
109. 

Index-numbers  83,  540  :  careless  re- 
jection of  189-90,  571n. 

Individual,  economic  89-94,  101, 
108-10,  118-20,  21 3n.,  301  n.;  in- 
different in  some  methods  192n., 
307-9,  359,  387n. 

Institut  international  de  Statistique, 
Reports  of  its  Cotnite  567n. 

Intervening  periods,  the  comparison 
should  pass  through  113n. 

Intrinsic:  of  (jualities  7n.,  12n.  ;  of 
values  1.33n. 

James,  H.,  his  method  537,  542  ;  B. 
10. 

Javolinus  137n. 

Jevons  6,  lln.,  22n  ,  .36n,  38n.,  41  n., 
81n.,  96n.,  158,  178n.,  180,  181n., 
184,  195,  220-1,  222n.,  223,  224, 
232,  253,  257,  355,  383,  432,  435, 
477,  478n.,  479n.,  493n.,  574  ;  his 
dispute  with  Laspeyres  220-1,  224- 
5,  2(56-9.  275,  354  and  n. ,  557, 
558,  559  ;'  B.   22-4. 

Jourdain  438  n. 

Kant  353n. 
Knies  136n. 
Krai  534,  536  ;  B.  98. 

Labor  :  not  being  exchangeable,  has 
has  not  exchange-value  122-4  ;  said 
to  be  the  real  price  of  things  4  and 
n.  ;  this  wrong,  it  is  the  ultimate 
cost  of  things  124-125  and  n. 

Laspevres98,  192n.,  196n.,  220,  221, 
222,"  225,  383,  47(5,  477n.,  54(i ;  the 
method  he  vised  84n.,  220,  536, 
545  ;  the  method  he  reconuuended 
99,  541,  550;  his  dispute  with 
Jevons,  see  Jevons  ;   B.  25-6. 

Laughlin  12n.,  56n.,  438n.,  478n. 

Laves,  T.  478  and  n.,  493n.,  495n. 

Leber  B.  12. 

Lehr  41n.,  86n.,  Ill,  182,  192n.,  197. 
198n.,  207n.,  222n.,  225,  393,  476. 
477n.,    544,  546,  571  ;  his   method 


0/; 


INDEX 


199,   204,  386-8,    391,   392n.,  410, 

417,  418  and  n.,  419n.,  420n.,  422, 

423,  427,  428,  547-8  ;  B.  68. 
Levasseur  41n.,   82n.,    137n.,  438n., 

534,  542  ;  B.  18. 
Lexis  25n. 
Lindsav  41n.,  113n.,  122n.,  541,  546, 

548  ;"B.  114. 
Locke  79. 
Logarithms  :    use   of,    in    averaging 

180-1,  220. 
Lowe  84,  477,  478  and  n.,  493n..  537  ; 

B.  8. 

McCnlloch  7n.,24n.,45n.,122n.,137n. 

Maclaren  478n. ;  B.  17. 

Maoleod  9n.,  lln.,  13-14n.,  29n., 
45n.,  56n.,  75n.,  438n. 

Malthus  5n.,  lln.,  24n.,  81n..  84n., 
126n.,  127n.,  135n.,  486-7. 

Mannequin  57n.,  137n.,  438n.,  439n., 
480n. 

Marshall  80n.,  84n.,  85n.,  97n.. 
113n.,  123n.,  222n.,  478n.,  495n., 
567  ;  B.  93. 

Martello  lOn.,  23n.,  45n.,  303. 

Martin  80n.,  122n.,  208n.,  567. 

Mass-unit :  employment  of  the  same 
in  all  classes  86,  90,98,  164-5,  170, 
190,  545,  547  ;  importance  of 
rightlv  selecting  89-90,  156,  162, 
182,  183,  202,  269,  276,  285ft:, 
344ff.,  550  ;  indifferent  in  some 
cases  182n.,  192n.,  307-10,  359, 
377,  387n.,  548. 

Mayo-Smith  208n.,  222n.;  B.  137-8. 

Mayr  567n. 

Mean  :  see  Average ;  median,  and  of 
greatest  thickness  224. 

Measure  of  value  :  confusion  in  Mal- 
thus's  24n.  ;  dispute  concerning, 
suggested  solution  r26n. 

Measurement  :  of  particular  ex- 
change-values 14-15  ;  involved  in 
the  conception  of  exchange-value 
21  ;  independent  of  causes  22-5  ;  of 
the  exchange-value  of  money  137, 
how  frequently  to  be  made  493n. ; 
principles  of,  with  constant  sums 
305,  with  constant  masses  359,  for 
all  cases  374-5. 

Measures  :  themselves  to  be  measured 
57  and  n.,  137. 

Menger,  C.  9-lOn. 

Mensuration,  simple  58. 

Messedaglia  223-4,  257,  383  ;  B.   52. 

Metrology  127. 

Mill,  J.  122n. 

Mill,  J.  S.  lln.,  13n.,  22n.,  29n., 
37n.,  41  n.,  438. 

Molinaeus  137n. 


Money  :  its  exchange-value  the  pri- 
mary one  to  be  measured  20-1  ;  de- 
nial of  this  rejected  137  ;  question 
whether  it  is  the  standard  of  ex- 
change-value, cost-value,  or  esteem- 
value  24,  135,  488-9,  495  ;  regula- 
tion of  its  quantity  with  a  view  to 
keeping  it  steady  in  exchange-value 
489-93,  or  in  cost-value  or  esteem- 
value  494  5. 

Money-value  5n. 

Montanari  137n. 

Mulhall  99n.,  541  ;  B.  74-75 

Multiple  standard  79n.,  473,  470,  cf. 
478. 

Nasse96n.,  122n.;  B.  103-4. 
Natural  :  of  price  4. 
Neumann-Spallart  567n. 
Xewcomb  122n.,  495n.,  540;  B.  76. 
Newmarch  536  ;  B.  19-21. 
Nicholson   22n.,  41n.,   113n.,   123n., 

143n.,   225,  544,  567;  his  method 

197,    204,    391,    427n.,    543,    544, 

548-51  ;  B.  94. 
Nitti  23n. 
Nominal  :  of  price  4  ;  of  value  4,  5 

and  n.,  lln. 

Objective  :  two  senses  of  85-6. 
O' Conor,  J.  E.  219n. 
Oker,  C.  W.  190n. 
Osborne,  G.  P.  122n.,  479n. 

Paasche  98,  99,  192n.,  222n.,  546 
and  n.;  his  varietv  of  Scrope's 
method  194,  197,  199,  204,  541; 
B.  33-4. 

Padan  190n.,  192n.,  222n.,  540,  544; 
B.  141. 

Palgrave  83,  84n.,  lOOn.,  219n.;  his 
varietv  of  Young's  method  194, 197, 
199,  204,  433,537,538-9,567;  B.  77. 

Pantaleoni  567  n. 

Parallelogram  of  forces:  none  in 
exchange-value  69n. 

Pareto  23n. 

Parsons  12n.,  122n.,  438n.,  478n., 
479n.,  493n.,  540;  B.  136. 

Patten  438n. 

Percentage :  mathematically  repre- 
sented in  hundredths  28 ;  three 
ways  of  reckoning  234  ft'. 

Periods :  question  of,  in  weighting 
97-121  ;  importance  of,  in  the 
treatment  of  variations  238  ;  gen- 
eral principles  concerning  244,  ap- 
plied 285  ft.,  302n.,  307,' 343,  344, 
348,  353. 

Perrv,  A.  L.  438n. 

Pieri^on,  N.  G.  190n.,  222n.;  B.  122 


IXDKX 


"Pleasure-unit"  886,  393,  547. 

"I'lutologv"  6n. 

Pollard,  T.  J.  127n.,  439n.,  494n. 

Pomerov  85n.,  493n.,  540;  B.  135. 

Porter,  'G.  K.  84n.,  208n.,  536;  B.  11. 

Powei-s,  L.  G.  ;i9n.,  536,  541,  544n., 
547  ;  B.  131. 

Preciousnes.-i :  defined  165n. ;  measure 
of,  of  commodities  200-1,  21  In. 

Price:  variously  used  4  and  n.,  29 
and  n. ;  distinfi;uislied  from  cost 
124-5  n.;  in  formuLe,  see  Formu- 
lation ;  variations  of,  and  variations 
of  exchange-value  482-3  ;  the  de- 
sire that  all  prices  should  fall  488. 

Price,  L.  L.,  B.  138. 

Prince-Smith  35n.,  36n.,  41n.,  45n., 
136n.,  143n. 

Prinsep,  ('.  ('.  219n. 

Probability  :  wrongly. invoked  where 
measurement  is  possible  38-9,  61, 
69. 

Propositions:  114;  II  15(33);  III 
16  ;  IV  16-17  ;  V  17  ;  VI  18  ;  VII 
18  (113,  114,  139,  141,  470);  VIII 
26  ;  IX  2S-9  (34n.,  UO,  447n. );  X 
29  (34n.,  174);  XI  29;  XII  30 
(471);  XIII  30  (140,  471);  XIV 
31  (48,  213);  XV  31  (447n.);  XVI 
32(447n.  ;  XVII  33  (42,  115n., 
149,  153.  196,  205n.,  212,  311,  361, 
388);  XVIII  34  (51,    212,  216n. ); 

XIX  34   (51,  212,  434,  455,  463); 

XX  36  (4(>,  150,  311,  361);  XXI 
41  (210);  XXII  42  (210,  457); 
XXIII  42  (210);  XXIV  42  (49, 
210,  45(),  4(33);  XXV  42  I  Hon., 
210,  211,  434,  456n.  );  XXVI  43 
(115n.,  210,  457);  XXVIl  44(115 
and  n.,  149,  153,  196,  205n.,  310, 
361,  385,  388.;  XXVIII  47  (150, 
311,361,447n.);XXIX47(447n.); 
XXX  48  (91,  92n.);  XXXI  48 
(434);  XXXII  48  (bOn.,  210, 
312n.,  .%ln.,  457);  XXXIII  48-49 
(69n.,  140,  214n.,  44t3n.,  453n., 
459n.,  471);  XXXIV  49  (453n.); 
XXXV  50  ( 210,  457 ) ;  XXXVI  50 
(180,  210,  311-12,  361,  394,  39(5); 
XXXVII  51  ;  XXXVIII  51  ; 
XXXIX  51  (438,  44.3,  446,  451); 
XL  52;  XLI  114;  XLII  114 
(440);  XLIII  114  (440);  XLIV 
115  (15(),  179,  196,  205n.,  310,  361, 
385,  388,  400);  XLV  115  (156,  179, 
196,  205n.,  311,  361,  388);  XLVI 
213  ;  XLVII  433  ;  XLVIII  435  ; 
XLIX  436  ;  L  437  ;  LI  441  ;  LII 
441  (485);  LIII  447;  LIV  449; 
LV  454;  LVI  456;  LVII  459 
(466,  471);   LVIII  467. 


Pufendorf  476  and  n. 
Purchase  :  meaning  173n. 
Purchasing  power  5,  7,  18n.,  ll-12n., 
30n.,  31n.,52,  172-3,  173n.,  303-5. 

Quantitativeness   of   excliange-value 

14-22. 
Quantity  :  ambiquity  of  the  term  86n. 

Real  :  of  price  4  ;  of  value  4,  5,  13n. ; 
of  exchange-value  13. 

Relative:  of  value  3n.,  5;  no  need 
of  the  epithet  before  exchange- 
value  39n.;  all  quantities  are  56, 
and  all  variations  of  quantities  6()- 
7. 

Ricardo4-5,  7n.,  23,  24n.,  25n.,  38n., 

'  S4n.,  12()n.,  135n.,  137n.,  439n 

Robertson,  J.   B.,  479n. 

Rogers,  J.  E.  T.  536  ;  B.  92. 

Ro.scher  45,  122n.,  225,  303,  546; 
B.  32. 

Rossi  45n.,  13f)n. 

Sauerbeck  84n.,  477n. ;  his  methods 
99n.,  536,  537,538,  541  ;  B.  79-90. 

Sav,  J.  B.,  5n.,  lln.,  22n.,  75n., 
79n. 

Schmid,  F.,  B.  106. 

Scrope  220,  477,  478n.,  495n.;  his 
method,  in  general  191,  203,  280, 
350,  352,  353,  360,  361,   363,   370, 

372,  382,  383,  396,  402,  408n., 
427-8,  430,  431,  443-5,  450,  534, 
539  44,  54(in.,  549,  556,  emended, 

373,  37(),  377,  395n.,  396  to  407, 
applied  to  arithmetic  means  99, 
37()n.,  377n.,  409-10,  413-23, 
424n.,  42()n.;  B.  9. 

Segnitz  97n. 

Senior,  N.  W.  438. 

Sergei,  C.  H.  534;  B.  120. 

Series  of  periods  :  how  to  lie  formed 
2(J6-7 ;  inconsistency  in  cross- 
measurements  203  I  see  Wester- 
gaard's  test)  ;  special  cases  avoid- 
ing such  inconsistency  204n.,  390- 
2,  397-8 ;  remedv,  unsatisfactorv 
334-6,  393,  398-9' 

Shadwell  12n.,  27n.,  127n.,  494n. 

Sidgwick  84n.,  99,  112,  11.3n.,  540, 
544,  567  ;   B.  56. 

Simon,  A.  432  ;  B.  72. 

Smith,  Adam  4,  5,  8n.,  lln.,  71hi., 
122n.,  124n.,  126n..  137n. 

Smith,  J.  A.  479n.,  540  ;  B.  129. 

Smith,  J.  B.,  55()n. 

Smith,  J.  P.  lOn.,  208n.,  536  ;  B.  7. 

Soetbeer  536,  571  ;  B.  14 -HI. 

Standard  :  sometimes  outside  57  ;  in 
simple    mensuration  nothing    uiti- 


080 


INDEX 


mate  58,  itself  a  relation  between  a 
whole  and  its  parts  58-1),  62,  03,  C}4, 
(50,  70,  in  neither  of  them  separately 
59 -()0  ;  tlie  whole  onh'  a  practicable 
one  62  ;  in  exchange-value,  should 
be  as  inclusive  as  possible  76-80. 

''Stfindard  of  desiderata"  85. 

Stewart,  D.  79n. 

Storch  79n. 

Subjective  :  two  senses  85-6. 

Taussig  85n.;  B.  121. 

Test  cases  324-41,  368-70,  423-33. 

"  Timiotology  "  6n. 

Todhunter  518n. 

Tooke  84n.,  556n. 

Torrens  23n. 

Transposition  :   argument  by  351-2  ; 

analysis  of  this  531-2. 
Trenholm  !ln.,  lln. 
True-price  466-8,  473-7. 
Turgot  3,  4,  124n.;  137n. 

Unit  of  exchange-value  13(). 

Use-value  :  described  1-6  ;  not  de- 
sirable that  things  should  fall  in 
485u. 

y^aleur :  apprecmtive  3n. ;  hhingeable 
3  ;  estimative  3. 

Value  :  defined  2-3  ;  four  kinds  of 
1-6  ;  these  ought  to  be  distin- 
guished by  special  epithets  1,  134  ; 
no  otiier  kinds  of  economic  128, 
Init  two  more  kinds  133n. ;  faulty 
uses  of  the  term  and  of  allied  terms 
36  and  n. 

Variations  :  arithmetic,  harmonic, 
and  geometric  235-6  ;  three  kinds 
(if,  witli  reference  to  zero  245-8  ; 
compensatory  524-30  ;  nature  of 
geometric  531. 

\'on  Jacob  84  n. 

Wages  :  not  to  be  counted  in  meas- 
uring variations  in  the  exchange- 
value  of  money  121-33;  wrong  use 
of,  alone,  in  measuring  esteem- 
value  126-7,  130,  494. 

Walker,  A.  534 ;  B.  27. 

Walker,  F.  .\.  8n.,  56n.,  1.37n.,  478n. 

Walra-s,  L.  6n.,  7n.,  lOn.,  lln.,27n., 
36n.,  45n.,  96n.,  ]38n.,  178n., 
IsOn.,  221-2,  223,  265n.,  432,  479, 
480n.,  4',l3n.,  540,  546n. ;  B.  69-72. 

Walsh,  H.  479n. 

Walsh,  K.  11.  478n.;  B.  13. 

Wasscrab  8  In.,  122n.,  537  ;  B.  105. 

Weight :  use  of  the  term  81  n. ;  of  the 
thing  whose  exchange-value  in  all 
otlier  things  is  bciuK  measured,  in- 


different 94-5  ;  of  classes  varying 
like  the  average,  indifferent  180n., 
500-1,  cf.  115n.;  of  wages  and 
of  commodities,  incommensurable 
132  ;  of  money  463-5. 

Weighting  :  defined  81  ;  explained 
87-89  ;  historv  of  84-7  ;  haphazard 
81-2,  156,  170,  188,  201,  534  ; 
even  82,  501  ;  {)roper  uneven,  need 
of  82-3  ;  rougli,  preferal)le  to  even 
83,  121,  431-2  ;  a  wrong  view  of 
85  ;  by  mass-quantities  86-7,  90, 
97,  165,  170,  190,  195;  by  custom- 
house returns  96n. ;  by  consump- 
tion or  production  95-6;  according 
to  relative  sizes  of  the  classes  81, 
501,  these  according  to  the  num- 
bers of  economic  individuals  in  them 
89,  94,  120-1  ;  question  of  periods, 
see  under  Periods ;  uneven,  in 
averaging  the  prices  of  each  article 
during  a  period  96-7  ;  even,  in 
averaging  the  weights  of  periods 
105,  386  7  ;  simple,  alone  possible 
in  averaging  variations  153  ;  hidden 
506  ;  perverted  161-6,  168,  186, 189, 
534,  535,  537  ;  double  98,  111,  195, 
225,  521-3,  544-52,  its  need  of  prop- 
erly selected  mass-units  156,  182, 
unintentionally  incurred  189,  193- 
4,  539,  541,  "551;  the  weight- 
ing needed  to  make  the  geometric 
average  good  239n.,  318n.,  408-9n. 

Wells,  U.  A.  24n. 

Westergaard  84n.,  179n.,  203,  206, 
222,  537  ;  his  test  205,  332n. ,  370, 
389,  391  n.,  393,  396,  398,  399,  402, 
409,  417,  537,  questioned  400-1; 
B.  110. 

Wetmore,  W.  S.  536,  570. 

Whatelev  6n. 

Whewell  23n. 

Whitehead  B.  119 

Whitelaw  479n.,  493n.,  536;  B.  130. 

\Vicksell99,  112,  123n.,  222n.,  332n., 
387n.,  540,  544,  548  ;   B.  139. 

Wiel)e  122n.,  536,  548  ;  B.  124. 

Will,  T.  E.  479n. 

Williams,  A.  479n.,  493n. 

Wilson,  W.  D.  123n. 

Winn,  II.    478n.,  479n. 

Wright,  C.  D.,  B.  73. 

Young,  Arthur  85,  97,  122n.,  132n., 
536;  his  method  194,  203,  204, 
383,  428,  429,  432,  433,  536-9, 
539-40,  555  ;  B.  6. 

Zuckerkandl  lUSn.,  478n.,  495n. ,  544, 
546,  548;   B.  115-16. 


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